Characterization of Subthreshold Voltage Fluctuations in Neuronal Membranes

Exact analysis of the subthreshold variability for conductance-based neuronal models with synchronous synaptic inputs

The spiking activity of neocortical neurons exhibits a striking level of variability, even when these networks are driven by identical stimuli. The approximately Poisson firing of neurons has led to the hypothesis that these neural networks operate in the asynchronous state. In the asynchronous state neurons fire independently from one another, so that the probability that a neuron experience synchronous synaptic inputs is exceedingly low. While the models of asynchronous neurons lead to observed spiking variability, it is not clear whether the asynchronous state can also account for the level of subthreshold membrane potential variability. We propose a new analytical framework to rigorously quantify the subthreshold variability of a single conductance-based neuron in response to synaptic inputs with prescribed degrees of synchrony. Technically we leverage the theory of exchangeability to model input synchrony via jump-process-based synaptic drives; we then perform a moment analysis of the stationary response of a neuronal model with all-or-none conductances that neglects post-spiking reset. As a result, we produce exact, interpretable closed forms for the first two stationary moments of the membrane voltage, with explicit dependence on the input synaptic numbers, strengths, and synchrony. For biophysically relevant parameters, we find that the asynchronous regime only yields realistic subthreshold variability (voltage variance ≅ 4-9mV 2 ) when driven by a restricted number of large synapses, compatible with strong thalamic drive. By contrast, we find that achieving realistic subthreshold variability with dense cortico-cortical inputs requires including weak but nonzero input synchrony, consistent with measured pairwise spiking correlations. We also show that without synchrony, the neural variability averages out to zero for all scaling limits with vanishing synaptic weights, independent of any balanced state hypothesis. This result challenges the theoretical basis for mean-field theories of the asynchronous state.

Dopamine-Mediated Major Depressive Disorder in the Neural Circuit of Ventral Tegmental Area-Nucleus Accumbens-Medial Prefrontal Cortex: From Biological Evidence to Computational Models

Major depressive disorder (MDD) is a serious psychiatric disorder, with an increasing incidence in recent years. The abnormal dopaminergic pathways of the midbrain cortical and limbic system are the key pathological regions of MDD, particularly the ventral tegmental area- nucleus accumbens- medial prefrontal cortex (VTA-NAc-mPFC) neural circuit. MDD usually occurs with the dysfunction of dopaminergic neurons in VTA, which decreases the dopamine concentration and metabolic rate in NAc/mPFC brain regions. However, it has not been fully explained how abnormal dopamine concentration levels affect this neural circuit dynamically through the modulations of ion channels and synaptic activities. We used Hodgkin-Huxley and dynamical receptor binding model to establish this network, which can quantitatively explain neural activity patterns observed in MDD with different dopamine concentrations by changing the kinetics of some ion channels. The simulation replicated some important pathological patterns of MDD at the level of neurons and circuits with low dopamine concentration, such as the decreased action potential frequency in pyramidal neurons of mPFC with significantly reduced burst firing frequency. The calculation results also revealed that NaP and KS channels of mPFC pyramidal neurons played key roles in the functional regulation of this neural circuit. In addition, we analyzed the synaptic currents and local field potentials to explain the mechanism of MDD from the perspective of dysfunction of excitation-inhibition balance, especially the disinhibition effect in the network. The significance of this article is that we built the first computational model to illuminate the effect of dopamine concentrations for the NAc-mPFC-VTA circuit between MDD and normal groups, which can be used to quantitatively explain the results of existing physiological experiments, predict the results for unperformed experiments and screen possible drug targets.

Emergence of Irregular Activity in Networks of Strongly Coupled Conductance-Based Neurons

Cortical neurons are characterized by irregular firing and a broad distribution of rates. The balanced state model explains these observations with a cancellation of mean excitatory and inhibitory currents, which makes fluctuations drive firing. In networks of neurons with current-based synapses, the balanced state emerges dynamically if coupling is strong, i.e., if the mean number of synapses per neuron K is large and synaptic efficacy is of the order of 1 / K . When synapses are conductance-based, current fluctuations are suppressed when coupling is strong, questioning the applicability of the balanced state idea to biological neural networks. We analyze networks of strongly coupled conductance-based neurons and show that asynchronous irregular activity and broad distributions of rates emerge if synaptic efficacy is of the order of 1/ log(K). In such networks, unlike in the standard balanced state model, current fluctuations are small and firing is maintained by a drift-diffusion balance. This balance emerges dynamically, without fine-tuning, if inputs are smaller than a critical value, which depends on synaptic time constants and coupling strength, and is significantly more robust to connection heterogeneities than the classical balanced state model. Our analysis makes experimentally testable predictions of how the network response properties should evolve as input increases.

Voltage Clamp Errors During Estimation of Concurrent Excitatory and Inhibitory Synaptic Input to Neurons with Dendrites

The whole-cell voltage clamp technique is commonly used to estimate synaptic conductances. While previous work has shown how these estimates are affected by series resistance and space clamp errors during isolated synaptic events, how voltage clamp errors impact on synaptic conductance estimates during concurrent excitation and inhibition is less clear. This issue is particularly relevant given that many studies now use the whole-cell voltage clamp technique to estimate synaptic conductances in vivo, where both excitation and inhibition are intact. Using both simplistic and morphologically realistic models, we investigate how imperfect voltage clamp conditions distort estimates of excitatory and inhibitory synaptic conductance estimated using the Borg-Graham method during concurrent synaptic input onto dendrites. These simulations demonstrate that dendritically located conductances are underestimated even when dynamic clamp reinjection faithfully reproduces the voltage response at the soma to the actual conductances. Inhibitory conductances are underestimated more than excitatory conductances, leading to errors in the excitatory to inhibitory conductance ratio and negative inhibitory conductance estimates during distal inhibition. Interactions between unclamped dendritic excitatory and inhibitory conductances also introduce correlations when the actual conductances are uncorrelated, as well as distortions in the time course of estimated excitatory and inhibitory conductances. Finally, we show that space clamp errors are exacerbated by the inclusion of dendritic voltage-activated conductances. In summary, we highlight issues with the interpretation of synaptic conductance estimates obtained using somatic whole-cell voltage clamp during concurrent excitatory and inhibitory input to neurons with dendrites.

Axonal Computations

Axons functionally link the somato-dendritic compartment to synaptic terminals. Structurally and functionally diverse, they accomplish a central role in determining the delays and reliability with which neuronal ensembles communicate. By combining their active and passive biophysical properties, they ensure a plethora of physiological computations. In this review, we revisit the biophysics of generation and propagation of electrical signals in the axon and their dynamics. We further place the computational abilities of axons in the context of intracellular and intercellular coupling. We discuss how, by means of sophisticated biophysical mechanisms, axons expand the repertoire of axonal computation, and thereby, of neural computation.

An Efficient Population Density Method for Modeling Neural Networks with Synaptic Dynamics Manifesting Finite Relaxation Time and Short-Term Plasticity

When incorporating more realistic synaptic dynamics, the computational efficiency of population density methods (PDMs) declines sharply due to the increase in the dimension of master equations. To avoid such a decline, we develop an efficient PDM, termed colored-synapse PDM (csPDM), in which the dimension of the master equations does not depend on the number of synapse-associated state variables in the underlying network model. Our goal is to allow the PDM to incorporate realistic synaptic dynamics that possesses not only finite relaxation time but also short-term plasticity (STP). The model equations of csPDM are derived based on the diffusion approximation on synaptic dynamics and probability density function methods for Langevin equations with colored noise. Numerical examples, given by simulations of the population dynamics of uncoupled exponential integrate-and-fire (EIF) neurons, show good agreement between the results of csPDM and Monte Carlo simulations (MCSs). Compared to the original full-dimensional PDM (fdPDM), the csPDM reveals more excellent computational efficiency because of the lower dimension of the master equations. In addition, it permits network dynamics to possess the short-term plastic characteristics inherited from plastic synapses. The novel csPDM has potential applicability to any spiking neuron models because of no assumptions on neuronal dynamics, and, more importantly, this is the first report of PDM to successfully encompass short-term facilitation/depression properties.

A discrete algebraic framework for stochastic systems which yield unique and exact solutions

Many physical systems exhibit random or stochastic components which shape or even drive their dynamic behavior. The stochastic models and equations describing such systems are typically assessed numerically, with a few exceptions allowing for a mathematically more rigorous treatment in the framework of stochastic calculus. However, even if exact solutions can be obtained in special cases, some results remain ambiguous due to the analytical foundation on which this calculus rests. In this work, we set out to identify the conceptual problem which renders stochastic calculus ambiguous, and exemplify a discrete algebraic framework which, for all practical intents and purposes, not just yields unique and exact solutions, but might also be capable of providing solutions to a much wider class of stochastic models.

Specific Relationship between the Shape of the Readiness Potential, Subjective Decision Time, and Waiting Time Predicted by an Accumulator Model with Temporally Autocorrelated Input Noise

Self-initiated movements are reliably preceded by a gradual buildup of neuronal activity known as the readiness potential (RP). Recent evidence suggests that the RP may reflect subthreshold stochastic fluctuations in neural activity that can be modeled as a process of accumulation to bound. One element of accumulator models that has been largely overlooked in the literature is the stochastic term, which is traditionally modeled as Gaussian white noise. While there may be practical reasons for this choice, we have long known that noise in neural systems is not white - it is long-term correlated with spectral density of the form 1/f^β(with roughly 1 < β < 3) across a broad range of spatial scales. I explored the behavior of a leaky stochastic accumulator when the noise over which it accumulates is temporally autocorrelated. I also allowed for the possibility that the RP, as measured at the scalp, might reflect the input to the accumulator (i.e., its stochastic noise component) rather than its output. These two premises led to two novel predictions that I empirically confirmed on behavioral and electroencephalography data from human subjects performing a self-initiated movement task. In addition to generating these two predictions, the model also suggested biologically plausible levels of autocorrelation, consistent with the degree of autocorrelation in our empirical data and in prior reports. These results expose new perspectives for accumulator models by suggesting that the spectral properties of the stochastic input should be allowed to vary, consistent with the nature of biological neural noise.

Coupling of synaptic inputs to local cortical activity differs among neurons and adapts after stimulus onset

Cortical activity contributes significantly to the high variability of sensory responses of interconnected pyramidal neurons, which has crucial implications for sensory coding. Yet, largely because of technical limitations of in vivo intracellular recordings, the coupling of a pyramidal neuron's synaptic inputs to the local cortical activity has evaded full understanding. Here we obtained excitatory synaptic conductance ( g) measurements from putative pyramidal neurons and local field potential (LFP) recordings from adjacent cortical circuits during visual processing in the turtle whole brain ex vivo preparation. We found a range of g-LFP coupling across neurons. Importantly, for a given neuron, g-LFP coupling increased at stimulus onset and then relaxed toward intermediate values during continued visual stimulation. A model network with clustered connectivity and synaptic depression reproduced both the diversity and the dynamics of g-LFP coupling. In conclusion, these results establish a rich dependence of single-neuron responses on anatomical, synaptic, and emergent network properties. NEW & NOTEWORTHY Cortical neurons are strongly influenced by the networks in which they are embedded. To understand sensory processing, we must identify the nature of this influence and its underlying mechanisms. Here we investigate synaptic inputs to cortical neurons, and the nearby local field potential, during visual processing. We find a range of neuron-to-network coupling across cortical neurons. This coupling is dynamically modulated during visual processing via biophysical and emergent network properties.

Network activity influences the subthreshold and spiking visual responses of pyramidal neurons in the three-layer turtle cortex

A primary goal of systems neuroscience is to understand cortical function, typically by studying spontaneous and stimulus-modulated cortical activity. Mounting evidence suggests a strong and complex relationship exists between the ongoing and stimulus-modulated cortical state. To date, most work in this area has been based on spiking in populations of neurons. While advantageous in many respects, this approach is limited in scope: it records the activity of a minority of neurons and gives no direct indication of the underlying subthreshold dynamics. Membrane potential recordings can fill these gaps in our understanding, but stable recordings are difficult to obtain in vivo. Here, we recorded subthreshold cortical visual responses in the ex vivo turtle eye-attached whole brain preparation, which is ideally suited for such a study. We found that, in the absence of visual stimulation, the network was "synchronous"; neurons displayed network-mediated transitions between hyperpolarized (Down) and depolarized (Up) membrane potential states. The prevalence of these slow-wave transitions varied across turtles and recording sessions. Visual stimulation evoked similar Up states, which were on average larger and less reliable when the ongoing state was more synchronous. Responses were muted when immediately preceded by large, spontaneous Up states. Evoked spiking was sparse, highly variable across trials, and mediated by concerted synaptic inputs that were, in general, only very weakly correlated with inputs to nearby neurons. Together, these results highlight the multiplexed influence of the cortical network on the spontaneous and sensory-evoked activity of individual cortical neurons.NEW & NOTEWORTHY Most studies of cortical activity focus on spikes. Subthreshold membrane potential recordings can provide complementary insight, but stable recordings are difficult to obtain in vivo. Here, we recorded the membrane potentials of cortical neurons during ongoing and visually evoked activity. We observed a strong relationship between network and single-neuron evoked activity spanning multiple temporal scales. The membrane potential perspective of cortical dynamics thus highlights the influence of intrinsic network properties on visual processing.

A Diffusion Approximation and Numerical Methods for Adaptive Neuron Models with Stochastic Inputs

Characterizing the spiking statistics of neurons receiving noisy synaptic input is a central problem in computational neuroscience. Monte Carlo approaches to this problem are computationally expensive and often fail to provide mechanistic insight. Thus, the field has seen the development of mathematical and numerical approaches, often relying on a Fokker-Planck formalism. These approaches force a compromise between biological realism, accuracy and computational efficiency. In this article we develop an extension of existing diffusion approximations to more accurately approximate the response of neurons with adaptation currents and noisy synaptic currents. The implementation refines existing numerical schemes for solving the associated Fokker-Planck equations to improve computationally efficiency and accuracy. Computer code implementing the developed algorithms is made available to the public.

Modularity Induced Gating and Delays in Neuronal Networks

Neural networks, despite their highly interconnected nature, exhibit distinctly localized and gated activation. Modularity, a distinctive feature of neural networks, has been recently proposed as an important parameter determining the manner by which networks support activity propagation. Here we use an engineered biological model, consisting of engineered rat cortical neurons, to study the role of modular topology in gating the activity between cell populations. We show that pairs of connected modules support conditional propagation (transmitting stronger bursts with higher probability), long delays and propagation asymmetry. Moreover, large modular networks manifest diverse patterns of both local and global activation. Blocking inhibition decreased activity diversity and replaced it with highly consistent transmission patterns. By independently controlling modularity and disinhibition, experimentally and in a model, we pose that modular topology is an important parameter affecting activation localization and is instrumental for population-level gating by disinhibition.

The capacity to transmit information between connected parts of a neuronal network is fundamental to its function. The organization of network connections (the topology of the network) is therefore expected to play an important role in determining network transmission. Since modular topology characterizes many brain circuits on multiple scales, investigating the role of modularity in activity gating is clearly desirable. By engineering such modular networks in vitro, we were able to perform such an investigation. Under these experimental conditions, we can independently control the degree of modularity, as well as inhibition in the network. We show that a combination of these two properties is highly beneficial from a communication perspective. Namely, it equips connected modules and large modular networks with the capacity to gate and temporally coordinate activity between the different parts of the network.

Epileptogenic networks and drug-resistant epilepsy: Present and future perspectives of epilepsy research-Utility for the epileptologist and the epilepsy surgeon

A multidisciplinary approach is required to understand the complex intricacies of drug-resistant epilepsy (DRE). A challenge that neurosurgeons across the world face is accurate localization of epileptogenic zone. A significant number of patients who have undergone resective brain surgery for epilepsy still continue to have seizures. The reason behind this therapy resistance still eludes us. Thus to develop a cure for the difficult to treat epilepsy, we need to comprehensively study epileptogenesis. Till date, most of the studies on DRE is focused on undermining the abnormal functioning of receptors involved in synaptic transmission and reduced levels of antiepileptic drugs around there targets. But recent advances in imaging and electrophysiological techniques have suggested the role epileptogenic networks in the process of epileptogenesis. According to this hypothesis, the local neurons recruit distant neurons through complex oscillatory circuits, which further recruit more distant neurons, thereby generating a hypersynchronus neuronal activity. The epileptogenic networks may be confined to the lesion or could propagate to distant focus. The success of surgery depends on the precision by which the epileptogenic network is determined while planning a surgical intervention. Here, we summarize various modalities of electrophysiological and imaging techniques to determine the functionally active epileptogenic networks. We also review evidence pertaining to the proposed role of epileptogenic network in abnormal synaptic transmission which is one of the major causes of epileptiform activity. Elucidation of current concepts in regulation of synaptic transmission by networks will help develop therapies for epilepsy cases that cannot be managed pharmacologically.

Sensory stimulation shifts visual cortex from synchronous to asynchronous states

In the mammalian cerebral cortex, neural responses are highly variable during spontaneous activity and sensory stimulation. To explain this variability, the cortex of alert animals has been proposed to be in an asynchronous high-conductance state in which irregular spiking arises from the convergence of large numbers of uncorrelated excitatory and inhibitory inputs onto individual neurons. Signatures of this state are that a neuron's membrane potential (Vm) hovers just below spike threshold, and its aggregate synaptic input is nearly Gaussian, arising from many uncorrelated inputs. Alternatively, irregular spiking could arise from infrequent correlated input events that elicit large fluctuations in Vm (refs 5, 6). To distinguish between these hypotheses, we developed a technique to perform whole-cell Vm measurements from the cortex of behaving monkeys, focusing on primary visual cortex (V1) of monkeys performing a visual fixation task. Here we show that, contrary to the predictions of an asynchronous state, mean Vm during fixation was far from threshold (14 mV) and spiking was triggered by occasional large spontaneous fluctuations. Distributions of Vm values were skewed beyond that expected for a range of Gaussian input, but were consistent with synaptic input arising from infrequent correlated events. Furthermore, spontaneous fluctuations in Vm were correlated with the surrounding network activity, as reflected in simultaneously recorded nearby local field potential. Visual stimulation, however, led to responses more consistent with an asynchronous state: mean Vm approached threshold, fluctuations became more Gaussian, and correlations between single neurons and the surrounding network were disrupted. These observations show that sensory drive can shift a common cortical circuitry from a synchronous to an asynchronous state.

Effects of noise on models of spiny dendrites

We study the effects of noise in two models of spiny dendrites. Through the introduction of different types of noise to both the Spike-diffuse-spike (SDS) and Baer-Rinzel (BR) models we investigate the change in behaviour of the travelling wave solution present in both deterministic systems, as noise intensity increases. We show that the speed of wave propagation in both the SDS and BR models respectively differs as the noise intensity in the spine heads increases. In contrast the cable is very robust to noise and as such the speed shows very little variation from the deterministic system. We introduce a space-dependent spine density, ρ(x), to the original Baer-Rinzel model and show how this modified model can mimic behaviour (under influence of noise) of both original systems, through variation of one parameter. We also show that the correlation time and length scales of the noise can enhance propagation of travelling wave solutions where the white noise dominates the underlying signal and produces noise induced phenomena.

Gain of spinal motoneurons measured from square and ramp current pulses

The gain of motoneurons (MNs) characterizes how variations in synaptic input are transformed in to variations in output firing and muscle contraction. Experimentally gain is often defined as the frequency-current relation observed in response to injected suprathreshold square current pulses or current ramps during intracellular recording. The gain of MNs is strongly affected by adaptation: transient gain in response to depolarization is usually higher than steady state gain measured during sustained depolarization. The transient and the stationary gain of neurons are separate entities that can be selectively modified. Here we investigated how the transient and the stationary gain of spinal MNs obtained from responses to square current pulses are related to gain estimated from the responses to the current ramps. We found, that the gain in response to current ramps is identical to the steady state gain during sustained depolarization. Therefore, gain modulation is more fully characterized with square current pulses than with current ramps.

Sensitivity of noisy neurons to coincident inputs

How do neurons compute? Two main theories compete: neurons could temporally integrate noisy inputs (rate-based theories) or they could detect coincident input spikes (spike timing-based theories). Correlations at fine timescales have been observed in many areas of the nervous system, but they might have a minor impact. To address this issue, we used a probabilistic approach to quantify the impact of coincidences on neuronal response in the presence of fluctuating synaptic activity. We found that when excitation and inhibition are balanced, as in the sensory cortex in vivo, synchrony in a very small proportion of inputs results in dramatic increases in output firing rate. Our theory was experimentally validated with in vitro recordings of cortical neurons of mice. We conclude that not only are noisy neurons well equipped to detect coincidences, but they are so sensitive to fine correlations that a rate-based description of neural computation is unlikely to be accurate in general.

Estimation of Time-Dependent Input from Neuronal Membrane Potential

The set of firing rates of the presynaptic excitatory and inhibitory neurons constitutes the input signal to the postsynaptic neuron. Estimation of the time-varying input rates from intracellularly recorded membrane potential is investigated here. For that purpose, the membrane potential dynamics must be specified. We consider the Ornstein-Uhlenbeck stochastic process, one of the most common single-neuron models, with time-dependent mean and variance. Assuming the slow variation of these two moments, it is possible to formulate the estimation problem by using a state-space model. We develop an algorithm that estimates the paths of the mean and variance of the input current by using the empirical Bayes approach. Then the input firing rates are directly available from the moments. The proposed method is applied to three simulated data examples: constant signal, sinusoidally modulated signal, and constant signal with a jump. For the constant signal, the estimation performance of the method is comparable to that of the traditionally applied maximum likelihood method. Further, the proposed method accurately estimates both continuous and discontinuous time-variable signals. In the case of the signal with a jump, which does not satisfy the assumption of slow variability, the robustness of the method is verified. It can be concluded that the method provides reliable estimates of the total input firing rates, which are not experimentally measurable.

Inhibitory "noise"

Cortical neurons in vivo may operate in high-conductance states, in which the major part of the neuron's input conductance is due to synaptic activity, sometimes several-fold larger than the resting conductance. We examine here the contribution of inhibition in such high-conductance states. At the level of the absolute conductance values, several studies have shown that cortical neurons in vivo are characterized by strong inhibitory conductances. However, conductances are balanced and spiking activity is mostly determined by fluctuations, but not much is known about excitatory and inhibitory contributions to these fluctuations. Models and dynamic-clamp experiments show that, during high-conductance states, spikes are mainly determined by fluctuations of inhibition, or by inhibitory “noise”. This stands in contrast to low-conductance states, in which excitatory conductances determine spiking activity. To determine these contributions from experimental data, maximum likelihood methods can be designed and applied to intracellular recordings in vivo. Such methods indicate that action potentials are indeed mostly correlated with inhibitory fluctuations in awake animals. These results argue for a determinant role for inhibitory fluctuations in evoking spikes, and do not support feed-forward modes of processing, for which opposite patterns are predicted.

Fokker-Planck description of conductance-based integrate-and-fire neuronal networks

Steady dynamics of coupled conductance-based integrate-and-fire neuronal networks in the limit of small fluctuations is studied via the equilibrium states of a Fokker-Planck equation. An asymptotic approximation for the membrane-potential probability density function is derived and the corresponding gain curves are found. Validity conditions are discussed for the Fokker-Planck description and verified via direct numerical simulations.

Characterizing neuronal activity by describing the membrane potential as a stochastic process

Cortical neurons behave similarly to stochastic processes, as a consequence of their irregularity and dense connectivity. Their firing pattern is close to a Poisson process, and their membrane potential (V(m)) is analogous to colored noise. One way to characterize this activity is to identify V(m) to a multidimensional stochastic process. We review here this approach and how it can be used to extract important statistical signatures of neuronal activity. The "VmD method" consists of fitting the V(m) distribution obtained intracellularly to analytic expressions derived from stochastic processes, and thereby deduce synaptic conductance parameters. However, this method requires at least two levels of V(m), which prevents applications to single-trial measurements. We also discuss methods that can be applied to single V(m) traces, such as power spectral analysis and the "STA method" to calculate spike-triggered average conductances based on a maximum likelihood procedure. A recently proposed method, the "VmT method", is based on the fusion of these two concepts. This method is analogous to the VmD method and estimates the mean excitatory and inhibitory conductances and their variances. However, it does so by using a maximum-likelihood estimation, and can thus be applied to single V(m) traces. All methods were tested using controlled conductance injection in dynamic-clamp experiments.

Subthreshold activation of spinal motoneurones in the stretch reflex: experimental data and modeling

Responses of gastrocnemius-soleus motoneurones to stretches of the homonymous muscles were recorded intrasomatically in decerebrate cats; changes of membrane potential (MP) were evoked by smoothed trapezoid stretches of the muscles. Amplitudes of separate excitatory postsynaptic potentials (EPSPs) were defined via differences between values of MP at the end and beginning of the positive derivative waves, which were also used as basic elements in the model of the excitatory postsynaptic currents (EPSCs). EPSCs were assumed to be transformed into EPSPs by low-pass filtering properties of the somatic membrane; parameters of the filtering were firstly defined from analysis of Ia EPSP in the same cell and then were applied in model P ( m0). The model showed unsatisfactory quality in tracking slow components of MP; to overcome the disadvantage there was proposed model P ( m1) based on addition to P ( m0) the difference between two low-pass filtered signals MP and P ( m0) (the cutoff frequency 10 or 20 Hz). An overestimation of EPSPs' amplitudes was corrected in model P ( m2). The mismatch in tracking slow changes of MP was assumed to be connected with summation of a great number of low-amplitude EPSPs generated at distal dendrites; information about waveform of separate EPSPs could disappear in this process. One can speculate that slow components of membrane depolarization at least partly are linked with the persistent inward currents in dendrites; variable and, sometimes, too fast decays in EPSPs seem to reflect inhibitory synaptic influences.

Stimulus-selective spiking is driven by the relative timing of synchronous excitation and disinhibition in cat striate neurons in vivo

What patterns of synaptic input cause cortical neurons to fire action potentials? Are they stochastic in nature, or do action potentials arise from the specific timing of synaptic input? We addressed these questions by measuring the membrane potential fluctuations associated with the generation of visually evoked action potentials in cat striate cortical neurons in vivo. In response to visual stimulation, action potentials occurred at the crest of large-amplitude, transient depolarizations (TDs) riding on sustained depolarization of the membrane potential. The magnitude, duration and rate of depolarization of these transient events were tuned for stimulus orientation. Using numerical simulations, we find that these transient events can arise from the temporal interplay between synchronous excitation and inhibition. To validate these findings, we made conductance measurements, at the preferred stimulus orientation, and showed that the TDs arise either from an increase in excitatory conductance, or from a combination of increased excitatory and decreased inhibitory conductance, both riding on sustained changes in synaptic conductances. The properties of the TDs and their underlying conductance suggest that they arise from a specific temporal interplay between synchronous excitatory and inhibitory synaptic inputs. Our results illustrate a mechanism by which the timing of synaptic inputs determines much of the spiking activity in striate cortical neurons.

High-resolution intracellular recordings using a real-time computational model of the electrode

Intracellular recordings of neuronal membrane potential are a central tool in neurophysiology. In many situations, especially in vivo, the traditional limitation of such recordings is the high electrode resistance and capacitance, which may cause significant measurement errors during current injection. We introduce a computer-aided technique, Active Electrode Compensation (AEC), based on a digital model of the electrode interfaced in real time with the electrophysiological setup. The characteristics of this model are first estimated using white noise current injection. The electrode and membrane contribution are digitally separated, and the recording is then made by online subtraction of the electrode contribution. Tests performed in vitro and in vivo demonstrate that AEC enables high-frequency recordings in demanding conditions, such as injection of conductance noise in dynamic-clamp mode, not feasible with a single high-resistance electrode until now. AEC should be particularly useful to characterize fast neuronal phenomena intracellularly in vivo.

Intrinsic gain modulation and adaptive neural coding

In many cases, the computation of a neural system can be reduced to a receptive field, or a set of linear filters, and a thresholding function, or gain curve, which determines the firing probability; this is known as a linear/nonlinear model. In some forms of sensory adaptation, these linear filters and gain curve adjust very rapidly to changes in the variance of a randomly varying driving input. An apparently similar but previously unrelated issue is the observation of gain control by background noise in cortical neurons: the slope of the firing rate versus current (f-I) curve changes with the variance of background random input. Here, we show a direct correspondence between these two observations by relating variance-dependent changes in the gain of f-I curves to characteristics of the changing empirical linear/nonlinear model obtained by sampling. In the case that the underlying system is fixed, we derive relationships relating the change of the gain with respect to both mean and variance with the receptive fields derived from reverse correlation on a white noise stimulus. Using two conductance-based model neurons that display distinct gain modulation properties through a simple change in parameters, we show that coding properties of both these models quantitatively satisfy the predicted relationships. Our results describe how both variance-dependent gain modulation and adaptive neural computation result from intrinsic nonlinearity.

Many neurons are known to achieve a wide dynamic range by adaptively changing their computational input/output function according to the input statistics. These adaptive changes can be very rapid, and it has been suggested that a component of this adaptation could be purely input-driven: even a fixed neural system can show apparent adaptive behavior since inputs with different statistics interact with the nonlinearity of the system in different ways. In this paper, we show how a single neuron's intrinsic computational function can dictate such input-driven changes in its response to varying input statistics, which begets a relationship between two different characterizations of neural function—in terms of mean firing rate and in terms of generating precise spike timing. We then apply our results to two biophysically defined model neurons, which have significantly different response patterns to inputs with various statistics. Our model of intrinsic adaptation explains their behaviors well. Contrary to the picture that neurons carry out a stereotyped computation on their inputs, our results show that even in the simplest cases they have simple yet effective mechanisms by which they can adapt to their input. Adaptation to stimulus statistics, therefore, is built into the most basic single neuron computations.

Two computational regimes of a single-compartment neuron separated by a planar boundary in conductance space

Recent in vitro data show that neurons respond to input variance with varying sensitivities. Here we demonstrate that Hodgkin-Huxley (HH) neurons can operate in two computational regimes: one that is more sensitive to input variance (differentiating) and one that is less sensitive (integrating). A boundary plane in the 3D conductance space separates these two regimes. For a reduced HH model, this plane can be derived analytically from the V nullcline, thus suggesting a means of relating biophysical parameters to neural computation by analyzing the neuron's dynamical system.

Characterizing synaptic conductance fluctuations in cortical neurons and their influence on spike generation

Cortical neurons are subject to sustained and irregular synaptic activity which causes important fluctuations of the membrane potential (V(m)). We review here different methods to characterize this activity and its impact on spike generation. The simplified, fluctuating point-conductance model of synaptic activity provides the starting point of a variety of methods for the analysis of intracellular V(m) recordings. In this model, the synaptic excitatory and inhibitory conductances are described by Gaussian-distributed stochastic variables, or "colored conductance noise". The matching of experimentally recorded V(m) distributions to an invertible theoretical expression derived from the model allows the extraction of parameters characterizing the synaptic conductance distributions. This analysis can be complemented by the matching of experimental V(m) power spectral densities (PSDs) to a theoretical template, even though the unexpected scaling properties of experimental PSDs limit the precision of this latter approach. Building on this stochastic characterization of synaptic activity, we also propose methods to qualitatively and quantitatively evaluate spike-triggered averages of synaptic time-courses preceding spikes. This analysis points to an essential role for synaptic conductance variance in determining spike times. The presented methods are evaluated using controlled conductance injection in cortical neurons in vitro with the dynamic-clamp technique. We review their applications to the analysis of in vivo intracellular recordings in cat association cortex, which suggest a predominant role for inhibition in determining both sub- and supra-threshold dynamics of cortical neurons embedded in active networks.

Spike-Frequency Adapting Neural Ensembles: Beyond Mean Adaptation and Renewal Theories

We propose a Markov process model for spike-frequency adapting neural ensembles that synthesizes existing mean-adaptation approaches, population density methods, and inhomogeneous renewal theory, resulting in a unified and tractable framework that goes beyond renewal and mean-adaptation theories by accounting for correlations between subsequent interspike intervals. A method for efficiently generating inhomogeneous realizations of the proposed Markov process is given, numerical methods for solving the population equation are presented, and an expression for the first-order interspike interval correlation is derived. Further, we show that the full five-dimensional master equation for a conductance-based integrate-and-fire neuron with spike-frequency adaptation and a relative refractory mechanism driven by Poisson spike trains can be reduced to a two-dimensional generalization of the proposed Markov process by an adiabatic elimination of fast variables. For static and dynamic stimulation, negative serial interspike interval correlations and transient population responses, respectively, of Monte Carlo simulations of the full five-dimensional system can be accurately described by the proposed two-dimensional Markov process.

Inhibition determines membrane potential dynamics and controls action potential generation in awake and sleeping cat cortex

Intracellular recordings of cortical neurons in awake cat and monkey show a depolarized state, sustained firing, and intense subthreshold synaptic activity. It is not known what conductance dynamics underlie such activity and how neurons process information in such highly stochastic states. Here, we combine intracellular recordings in awake and naturally sleeping cats with computational models to investigate subthreshold dynamics of conductances and how conductance dynamics determine spiking activity. We show that during both wakefulness and the "up-states" of natural slow-wave sleep, membrane-potential activity stems from a diversity of combinations of excitatory and inhibitory synaptic conductances, with dominant inhibition in most of the cases. Inhibition also provides the largest contribution to membrane potential fluctuations. Computational models predict that in such inhibition-dominant states, spikes are preferentially evoked by a drop of inhibitory conductance, and that its signature is a transient drop of membrane conductance before the spike. This pattern of conductance change is indeed observed in estimates of spike-triggered averages of synaptic conductances during wakefulness and slow-wave sleep up states. These results show that activated states are defined by diverse combinations of excitatory and inhibitory conductances with pronounced inhibition, and that the dynamics of inhibition is particularly effective on spiking, suggesting an important role for inhibitory processes in both conscious and unconscious cortical states.

Analysis of a stochastic neuronal model with excitatory inputs and state-dependent effects

We propose a stochastic model for the firing activity of a neuronal unit. It includes the decay effect of the membrane potential in absence of stimuli, and the occurrence of time-varying excitatory inputs governed by a Poisson process. The sample-paths of the membrane potential are piecewise exponentially decaying curves with jumps of random amplitudes occurring at the input times. An analysis of the probability distributions of the membrane potential and of the firing time is performed. In the special case of time-homogeneous stimuli the firing density is obtained in closed form, together with its mean and variance.

Non-Gaussian membrane potential dynamics imply sparse, synchronous activity in auditory cortex

Many models of cortical dynamics have focused on the high-firing regime, in which neurons are driven near their maximal rate. Here we consider the responses of neurons in auditory cortex under typical low-firing rate conditions, when stimuli have not been optimized to drive neurons maximally. We used whole-cell patch-clamp recording in vivo to measure subthreshold membrane potential fluctuations in rat primary auditory cortex in both the anesthetized and awake preparations. By analyzing the subthreshold membrane potential dynamics on single trials, we made inferences about the underlying population activity. We found that, during both spontaneous and evoked responses, membrane potential was highly non-Gaussian, with dynamics consisting of occasional large excursions (sometimes tens of millivolts), much larger than the small fluctuations predicted by most random walk models that predict a Gaussian distribution of membrane potential. Thus, presynaptic inputs under these conditions are organized into quiescent periods punctuated by brief highly synchronous volleys, or "bumps." These bumps were typically so brief that they could not be well characterized as "up states" or "down states." We estimate that hundreds, perhaps thousands, of presynaptic neurons participate in the largest volleys. These dynamics suggest a computational scheme in which spike timing is controlled by concerted firing among input neurons rather than by small fluctuations in a sea of background activity.

On the Use of Analytical Expressions for the Voltage Distribution to Analyze Intracellular Recordings

Different analytical expressions for the membrane potential distribution of membranes subject to synaptic noise have been proposed and can be very helpful in analyzing experimental data. However, all of these expressions are either approximations or limit cases, and it is not clear how they compare and which expression should be used in a given situation. In this note, we provide a comparison of the different approximations available, with an aim of delineating which expression is most suitable for analyzing experimental data.

Estimation of synaptic conductances

In order to identify and understand mechanistically the cortical circuitry of sensory information processing estimates are needed of synaptic input fields that drive neurons. From intracellular in vivo recordings one would like to estimate net synaptic conductance time courses for excitation and inhibition, g(E)(t) and g(I)(t), during time-varying stimulus presentations. However, the intrinsic conductance transients associated with neuronal spiking can confound such estimates, and thereby jeopardize functional interpretations. Here, using a conductance-based pyramidal neuron model we illustrate errors in estimates when the influence of spike-generating conductances are not reduced or avoided. A typical estimation procedure involves approximating the current-voltage relation at each time point during repeated stimuli. The repeated presentations are done in a few sets, each with a different steady bias current. From the trial-averaged smoothed membrane potential one estimates total membrane conductance and then dissects out estimates for g(E)(t) and g(I)(t). Simulations show that estimates obtained during phases without spikes are good but those obtained from phases with spiking should be viewed with skeptism. For the simulations, we consider two different synaptic input scenarios, each corresponding to computational network models of orientation tuning in visual cortex. One input scenario mimics a push-pull arrangement for g(E)(t) and g(I)(t) and idealized as specified smooth time courses. The other is taken directly from a large-scale network simulation of stochastically spiking neurons in a slab of cortex with recurrent excitation and inhibition. For both, we show that spike-generating conductances cause serious errors in the estimates of g(E) and g(I). In some phases for the push-pull examples even the polarity of g(I) is mis-estimated, indicating significant increase when g(I) is actually decreased. Our primary message is to be cautious about forming interpretations based on estimates developed during spiking phases.

Comment on "Characterization of subthreshold voltage fluctuations in neuronal membranes," by M. Rudolph and A. Destexhe

In two recent articles, Rudolph and Destexhe (2003, 2005) studied a leaky integrator model (an RC-circuit) driven by correlated ("colored") gaussian conductance noise and Gaussian current noise. In the first article, they derived an expression for the stationary probability density of the membrane voltage; in the second, they modified this expression to cover a larger parameter regime. Here we show by standard analysis of solvable limit cases (white noise limit of additive and multiplicative noise sources; only slow multiplicative noise; only additive noise) and by numerical simulations that their first result does not hold for the general colored-noise case and uncover the errors made in the derivation of a Fokker-Planck equation for the probability density. Furthermore, we demonstrate analytically (including an exact integral expression for the time-dependent mean value of the voltage) and by comparison to simulation results that the extended expression for the probability density works much better but still does not exactly solve the full colored-noise problem. We also show that at stronger synaptic input, the stationary mean value of the linear voltage model may diverge and give an exact condition relating the system parameters for which this takes place.

Orientation selectivity in visual cortex by fluctuation-controlled criticality

Within a large-scale neuronal network model of macaque primary visual cortex, we examined how intrinsic dynamic fluctuations in synaptic currents modify the effect of strong recurrent excitation on orientation selectivity. Previously, we showed that, using a strong network inhibition countered by feedforward and recurrent excitation, the cortical model reproduced many observed properties of simple and complex cells. However, that network's complex cells were poorly selective for orientation, and increasing cortical self-excitation led to network instabilities and unrealistically high firing rates. Here, we show that a sparsity of connections in the network produces large, intrinsic fluctuations in the cortico-cortical conductances that can stabilize the network and that there is a critical level of fluctuations (controllable by sparsity) that allows strong cortical gain and the emergence of orientation-selective complex cells. The resultant sparse network also shows near contrast invariance in its selectivity and, in agreement with recent experiments, has extracellular tuning properties that are similar in pinwheel center and iso-orientation regions, whereas intracellular conductances show positional dependencies. Varying the strength of synaptic fluctuations by adjusting the sparsity of network connectivity, we identified a transition between the dynamics of bistability and without bistability. In a network with strong recurrent excitation, this transition is characterized by a near hysteretic behavior and a rapid rise of network firing rates as the synaptic drive or stimulus input is increased. We discuss the connection between this transition and orientation selectivity in our model of primary visual cortex.

Fast numerical methods for simulating large-scale integrate-and-fire neuronal networks

We discuss numerical methods for simulating large-scale, integrate-and-fire (I&F) neuronal networks. Important elements in our numerical methods are (i) a neurophysiologically inspired integrating factor which casts the solution as a numerically tractable integral equation, and allows us to obtain stable and accurate individual neuronal trajectories (i.e., voltage and conductance time-courses) even when the I&F neuronal equations are stiff, such as in strongly fluctuating, high-conductance states; (ii) an iterated process of spike-spike corrections within groups of strongly coupled neurons to account for spike-spike interactions within a single large numerical time-step; and (iii) a clustering procedure of firing events in the network to take advantage of localized architectures, such as spatial scales of strong local interactions, which are often present in large-scale computational models-for example, those of the primary visual cortex. (We note that the spike-spike corrections in our methods are more involved than the correction of single neuron spike-time via a polynomial interpolation as in the modified Runge-Kutta methods commonly used in simulations of I&F neuronal networks.) Our methods can evolve networks with relatively strong local interactions in an asymptotically optimal way such that each neuron fires approximately once in [Formula: see text] operations, where N is the number of neurons in the system. We note that quantifications used in computational modeling are often statistical, since measurements in a real experiment to characterize physiological systems are typically statistical, such as firing rate, interspike interval distributions, and spike-triggered voltage distributions. We emphasize that it takes much less computational effort to resolve statistical properties of certain I&F neuronal networks than to fully resolve trajectories of each and every neuron within the system. For networks operating in realistic dynamical regimes, such as strongly fluctuating, high-conductance states, our methods are designed to achieve statistical accuracy when very large time-steps are used. Moreover, our methods can also achieve trajectory-wise accuracy when small time-steps are used.

Estimation of the input parameters in the Feller neuronal model

The stochastic Feller neuronal model is studied, and estimators of the model input parameters, depending on the firing regime of the process, are derived. Closed expressions for the first two moments of functionals of the first-passage time (FTP) through a constant boundary in the suprathreshold regime are derived, which are used to calculate moment estimators. In the subthreshold regime, the exponentiality of the FTP is utilized to characterize the input parameters. The methods are illustrated on simulated data. Finally, approximations of the first-passage-time moments are suggested, and biological interpretations and comparisons of the parameters in the Feller and the Ornstein-Uhlenbeck models are discussed.

Synaptic input statistics tune the variability and reproducibility of neuronal responses

Synaptic waveforms, constructed from excitatory and inhibitory presynaptic Poisson trains, are presented to living and computational neurons. We review how the average output of a neuron (e.g., the firing rate) is set by the difference between excitatory and inhibitory event rates while neuronal variability is set by their sum. We distinguish neuronal variability from reproducibility. Variability quantifies how much an output measure is expected to vary; for example, the interspike interval coefficient of variation quantifies the typical range of interspike intervals. Reproducibility quantifies the similarity of neuronal outputs in response to repeated presentations of identical stimuli. Although variability and reproducibility are conceptually distinct, we show that, for ideal current source synapses, reproducibility is defined entirely by variability. For physiologically realistic conductance-based synapses, however, reproducibility is distinct from variability and average output, set by the Poisson rate and the degree of synchrony within the synaptic waveform.

Statistics of subthreshold neuronal voltage fluctuations due to conductance-based synaptic shot noise

Neurons in the central nervous system, and in the cortex in particular, are subject to a barrage of pulses from their presynaptic populations. These synaptic pulses are mediated by conductance changes and therefore lead to increases or decreases of the neuronal membrane potential with amplitudes that are dependent on the voltage: synaptic noise is multiplicative. The statistics of the membrane potential are of experimental interest because the measurement of a single subthreshold voltage can be used to probe the activity occurring across the presynaptic population. Though the interpulse interval is not always significantly smaller than the characteristic decay time of the pulses, and so the fluctuations have the nature of shot noise, the majority of results available in the literature have been calculated in the diffusion limit, which is valid for high-rate pulses. Here the effects that multiplicative conductance noise and shot noise have on the voltage fluctuations are examined. It is shown that both these aspects of synaptic drive sculpt high-order features of the subthreshold voltage distribution, such as the skew. It is further shown that the diffusion approximation can only capture the effects arising from the multiplicative conductance noise, predicting a negative voltage skew for excitatory drive. Exact results for the full dynamics are derived from a master-equation approach, predicting positively skewed distributions with long tails in voltage ranges typical for action potential generation. It is argued that, although the skew is a high-order feature of subthreshold voltage distributions, the increased probability of reaching firing threshold suggests a potential role for shot noise in shaping the neuronal transfer function.

Efficient estimation of detailed single-neuron models

Biophysically accurate multicompartmental models of individual neurons have significantly advanced our understanding of the input-output function of single cells. These models depend on a large number of parameters that are difficult to estimate. In practice, they are often hand-tuned to match measured physiological behaviors, thus raising questions of identifiability and interpretability. We propose a statistical approach to the automatic estimation of various biologically relevant parameters, including 1) the distribution of channel densities, 2) the spatiotemporal pattern of synaptic input, and 3) axial resistances across extended dendrites. Recent experimental advances, notably in voltage-sensitive imaging, motivate us to assume access to: i) the spatiotemporal voltage signal in the dendrite and ii) an approximate description of the channel kinetics of interest. We show here that, given i and ii, parameters 1-3 can be inferred simultaneously by nonnegative linear regression; that this optimization problem possesses a unique solution and is guaranteed to converge despite the large number of parameters and their complex nonlinear interaction; and that standard optimization algorithms efficiently reach this optimum with modest computational and data requirements. We demonstrate that the method leads to accurate estimations on a wide variety of challenging model data sets that include up to about 10(4) parameters (roughly two orders of magnitude more than previously feasible) and describe how the method gives insights into the functional interaction of groups of channels.

Synaptic bombardment modulates muscarinic effects in forelimb motor cortex

Neocortical neurons in vivo exist in an environment of continuous synaptic bombardment, receiving a complex barrage of excitatory and inhibitory inputs. This background activity (by depolarizing neurons, increasing membrane conductance, and introducing fluctuations) strongly alters many aspects of neuronal responsiveness. In this study, we asked how it shapes neuromodulation of postsynaptic responses. Specifically, we examined muscarinic modulation of forelimb motor cortex, a brain area in which cholinergic stimulation is known to be necessary for modifications during motor skill learning. Using a dynamic clamp system to inject simulated conductances into pyramidal neurons in motor cortical slices, we mimicked in vivo-like activity by introducing a random background of excitatory and inhibitory inputs. When muscarinic receptors were stimulated with the agonist oxotremorine-M, several previously described currents were modified, and excitability was increased. However, the presence of the background conductances strongly attenuated most muscarinic agonist effects, with the notable exception that sustained firing responses to trains of inputs were well preserved. This may be important for promoting plasticity in vivo.

Sustained attention and apical dendrite activity in recurrent circuits

Recurrent neural activity is a pervasive mode of cortical operations and is believed to underlie cognitive functions of working memory, attention, and the generation of spontaneous activity during sleep . It is proposed here that activity in corticothalamic recurrent circuits underlies the sustaining of attention, and that extended durations of attention are made possible by the stabilizing effects of electrical activity in long apical dendrites of pyramidal neurons. Using the cue-target delay task as a framework, the present paper describes sustained attention during the cue-target delay as activity in recurrent circuits involving layer 5/6 pyramidal neurons. At target onset, persistent activity in apical dendrites of layer 2/3 pyramidal neurons (projected from the recurrent circuits) can enhance the processing of incoming pulse trains at basal dendrites. Apical dendrite activity is assumed to modulate the soma processing of layer 2/3 and layer 5/6 pyramidal neurons at subthreshold voltage levels. The variability of successive soma depolarizations from the apical dendrite strongly influences the stability of activity in the corticothalamic recurrent circuit. Lower variability promotes higher stability. According to the present model of apical dendrite function, soma depolarizations can be reduced in variability and maintained within subthreshold levels by increasing the distance that EPSPs propagate along the apical dendrite. The close relationship between sustained attention and the electrical field potentials produced by repeated EPSP propagations in apical dendrites is supported in a brief review of sustained attention experiments that have employed measures of EEG, ERS/ERD, ERP, and LFP.

Periodic high-conductance states in spinal neurons during scratch-like network activity in adult turtles

Intense synaptic activity may alter the response properties of neurons in highly interconnected networks. Here we investigate whether the excitability and the intrinsic response properties of neurons in the spinal cord are affected by the increased synaptic conductance during functional network activity. Scratch episodes were induced by mechanical stimulation in the isolated carapace-spinal cord preparation from the adult turtle. Intracellular recordings revealed a dramatic increase in synaptic activity in interneurons and motoneurons during scratch activity. Superimposed slow depolarizing waves were phase-related to the rhythmic bouts of spike activity in the hip flexor nerve. The increase in synaptic conductance in interneurons and motoneurons varied with the scratch rhythm. During individual episodes, the conductance shifted smoothly with the scratch rhythm from near-resting levels to levels two to four times higher. In slice experiments, we found that even moderate increases in the conductance of motoneurons suppressed the slow afterhyperpolarization and the plateau potentials. We conclude that the excitability and the intrinsic response properties of spinal neurons are periodically quenched by high synaptic conductance during functional network activity.

Relation between single neuron and population spiking statistics and effects on network activity

To simplify theoretical analyses of neural networks, individual neurons are often modeled as Poisson processes. An implicit assumption is that even if the spiking activity of each neuron is non-Poissonian, the composite activity obtained by summing many spike trains limits to a Poisson process. Here, we show analytically and through simulations that this assumption is invalid. Moreover, we show with Fokker-Planck equations that the behavior of feedforward networks is reproduced accurately only if the tendency of neurons to fire periodically is incorporated by using colored noise whose autocorrelation has a negative component.

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.

Predicting spike times of a detailed conductance-based neuron model driven by stochastic spike arrival

Reduced models of neuronal activity such as integrate-and-fire models allow a description of neuronal dynamics in simple, intuitive terms and are easy to simulate numerically. We present a method to fit an integrate-and-fire-type model of neuronal activity, namely a modified version of the spike response model, to a detailed Hodgkin-Huxley-type neuron model driven by stochastic spike arrival. In the Hogkin-Huxley model, spike arrival at the synapse is modeled by a change of synaptic conductance. For such conductance spike input, more than 70% of the postsynaptic action potentials can be predicted with the correct timing by the integrate-and-fire-type model. The modified spike response model is based upon a linearized theory of conductance-driven integrate-and-fire neurons.

An Extended Analytic Expression for the Membrane Potential Distribution of Conductance-Based Synaptic Noise

Synaptically generated subthreshold membrane potential (Vm) fluctuations can be characterized within the framework of stochastic calculus. It is possible to obtain analytic expressions for the steady-state Vm 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 Vm 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 Vm distributions and may be applicable to other classes of stochastic processes as well.