Membrane Resonance and Stochastic Resonance Modulate Firing Patterns of Thalamocortical Neurons

We examined the interactions of subthreshold membrane resonance and stochastic resonance using whole-cell patch clamp recordings in thalamocortical neurons of rat brain slices, as well as with a Hodgkin-Huxley-type mathematical model of thalamocortical neurons. The neurons exhibited the subthreshold resonance when stimulated with small amplitude sine wave currents of varying frequency, and stochastic resonance when noise was added to sine wave inputs. Stochastic resonance was manifest as a maximum in signal-to-noise ratio of output response to subthreshold periodic input combined with noise. Stochastic resonance in conjunction with subthreshold resonance resulted in action potential patterns that showed frequency selectivity for periodic inputs. Stochastic resonance was maximal near subthreshold resonance frequency and a high noise level was required for detection of high frequency signals. We speculate that combined membrane and stochastic resonances have physiological utility in coupling synaptic activity to preferred firing frequency and in network synchronization under noise.     

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.     

Balanced Excitatory and Inhibitory Synaptic Currents Promote Efficient Coding and Metabolic Efficiency

A balance between excitatory and inhibitory synaptic currents is thought to be important for several aspects of information processing in cortical neurons in vivo, including gain control, bandwidth and receptive field structure. These factors will affect the firing rate of cortical neurons and their reliability, with consequences for their information coding and energy consumption. Yet how balanced synaptic currents contribute to the coding efficiency and energy efficiency of cortical neurons remains unclear. We used single compartment computational models with stochastic voltage-gated ion channels to determine whether synaptic regimes that produce balanced excitatory and inhibitory currents have specific advantages over other input regimes. Specifically, we compared models with only excitatory synaptic inputs to those with equal excitatory and inhibitory conductances, and stronger inhibitory than excitatory conductances (i.e. approximately balanced synaptic currents). Using these models, we show that balanced synaptic currents evoke fewer spikes per second than excitatory inputs alone or equal excitatory and inhibitory conductances. However, spikes evoked by balanced synaptic inputs are more informative (bits/spike), so that spike trains evoked by all three regimes have similar information rates (bits/s). Consequently, because spikes dominate the energy consumption of our computational models, approximately balanced synaptic currents are also more energy efficient than other synaptic regimes. Thus, by producing fewer, more informative spikes approximately balanced synaptic currents in cortical neurons can promote both coding efficiency and energy efficiency.     

Power Laws from Linear Neuronal Cable Theory: Power Spectral Densities of the Soma Potential,Soma Membrane Current and Single-Neuron Contribution to the EEG

Power laws, that is, power spectral densities (PSDs) exhibiting behavior for large frequencies f, have been observed both in microscopic (neural membrane potentials and currents) and macroscopic (electroencephalography; EEG) recordings. While complex network behavior has been suggested to be at the root of this phenomenon, we here demonstrate a possible origin of such power laws in the biophysical properties of single neurons described by the standard cable equation. Taking advantage of the analytical tractability of the so called ball and stick neuron model, we derive general expressions for the PSD transfer functions for a set of measures of neuronal activity: the soma membrane current, the current-dipole moment (corresponding to the single-neuron EEG contribution), and the soma membrane potential. These PSD transfer functions relate the PSDs of the respective measurements to the PSDs of the noisy input currents. With homogeneously distributed input currents across the neuronal membrane we find that all PSD transfer functions express asymptotic high-frequency power laws with power-law exponents analytically identified as for the soma membrane current, for the current-dipole moment, and for the soma membrane potential. Comparison with available data suggests that the apparent power laws observed in the high-frequency end of the PSD spectra may stem from uncorrelated current sources which are homogeneously distributed across the neural membranes and themselves exhibit pink () noise distributions. While the PSD noise spectra at low frequencies may be dominated by synaptic noise, our findings suggest that the high-frequency power laws may originate in noise from intrinsic ion channels. The significance of this finding goes beyond neuroscience as it demonstrates how power laws with a wide range of values for the power-law exponent α may arise from a simple, linear partial differential equation.     

Comparison of Coding Capabilities of Type I and Type II Neurons

We consider the dependence of information transfer by neurons on the Type I vs. Type II classification of their dynamics. Our computational study is based on Type I and II implementations of the Morris-Lecar model. It mainly concerns neurons, such as those in the auditory or electrosensory system, which encode band-limited amplitude modulations of a periodic carrier signal, and which fire at random cycles yet preferred phases of this carrier. We first show that the Morris-Lecar model with additive broadband noise ("synaptic noise") can exhibit such firing patterns with either Type I or II dynamics, with or without amplitude modulations of the carrier. We then compare the encoding of band-limited random amplitude modulations for both dynamical types. The comparison relies on a parameter calibration that closely matches firing rates for both models across a range of parameters. In the absence of synaptic noise, Type I performs slightly better than Type II, and its performance is optimal for perithreshold signals. However, Type II performs well over a slightly larger range of inputs, and this range lies mostly in the subthreshold region. Further, Type II performs marginally better than Type I when synaptic noise, which yields more realistic baseline firing patterns, is present in both models. These results are discussed in terms of the tuning and phase locking properties of the models with deterministic and stochastic inputs.     

The generation of direction selectivity in the auditory system

Both human speech and animal vocal signals contain frequency-modulated (FM) sounds. Although central auditory neurons that selectively respond to the direction of frequency modulation are known, the synaptic mechanisms underlying the generation of direction selectivity (DS) remain elusive. Here we show the emergence of DS neurons in the inferior colliculus by mapping the three major subcortical auditory nuclei. Cell-attached recordings reveal a highly reliable and precise firing of DS neurons to FM sweeps in a preferred direction. By using in vivo whole-cell current-clamp and voltage-clamp recordings, we found that the synaptic inputs to DS neurons are not direction selective, but temporally reversed excitatory and inhibitory synaptic inputs are evoked in response to opposing directions of FM sweeps. The construction of such temporal asymmetry, resulting DS, and its topography can be attributed to the spectral disparity of the excitatory and the inhibitory synaptic tonal receptive fields.     

Spike timing precision of neuronal circuits

Spike timing is believed to be a key factor in sensory information encoding and computations performed by the neurons and neuronal circuits. However, the considerable noise and variability, arising from the inherently stochastic mechanisms that exist in the neurons and the synapses, degrade spike timing precision. Computational modeling can help decipher the mechanisms utilized by the neuronal circuits in order to regulate timing precision. In this paper, we utilize semi-analytical techniques, which were adapted from previously developed methods for electronic circuits, for the stochastic characterization of neuronal circuits. These techniques, which are orders of magnitude faster than traditional Monte Carlo type simulations, can be used to directly compute the spike timing jitter variance, power spectral densities, correlation functions, and other stochastic characterizations of neuronal circuit operation. We consider three distinct neuronal circuit motifs: Feedback inhibition, synaptic integration, and synaptic coupling. First, we show that both the spike timing precision and the energy efficiency of a spiking neuron are improved with feedback inhibition. We unveil the underlying mechanism through which this is achieved. Then, we demonstrate that a neuron can improve on the timing precision of its synaptic inputs, coming from multiple sources, via synaptic integration: The phase of the output spikes of the integrator neuron has the same variance as that of the sample average of the phases of its inputs. Finally, we reveal that weak synaptic coupling among neurons, in a fully connected network, enables them to behave like a single neuron with a larger membrane area, resulting in an improvement in the timing precision through cooperation.     

Central generation of swimming activity in the hydrozoan jellyfish Aequorea aequorea

Swimming in Aequorea is controlled by a network of electrically coupled neurons (swim motorneurons) located in the inner nerve ring. The network is made up of the largest neurons in the ring, up to 22 microns in diameter. Intracellular recordings from swim motorneurons reveal slow membrane potential oscillations and a superimposed barrage of synaptic "noise." The synaptic noise, but not the slow oscillations, is eliminated in seawater containing an elevated Mg++ concentration. The swim motorneurons produce a rapid burst of two to eight action potentials preceding each contraction of the subumbrella. Spontaneous bursting persists in high-Mg++ seawater. Injected ramp currents indicated a "bursty" character of the swim motorneurons as suprathreshold depolarizations produced repetitive bursting with an increasing burst frequency with increased depolarization. Hyperpolarizing currents locally blocked spiking in swim motorneurons. Intercellular coupling was demonstrated with Lucifer Yellow injection and dual electrode recordings. In dye fills, only the large neurons of the inner nerve ring were dye-coupled. Two pieces of evidence suggest that swim motorneurons activate the overlying epithelial cells via chemical synapses. First, direct synaptic connections have been noted in ultrastructural examination of the inner nerve ring region. Second, dual recordings from a swim motorneuron and an epithelial cell reveal a 1:1 correspondence between neuron spikes and epithelial synaptic potentials. The synaptic potentials occur with a latency as short as 3 ms which is constant in any one recording session. The results suggest that the swim motorneuron network of Aequorea not only performs a motorneuron function, but also serves as the pattern generator for swimming activity.     

Analytical and Simulation Results for Stochastic Fitzhugh-Nagumo Neurons and Neural Networks

An analytical approach is presented for determining the response of a neuron or of the activity in a network of connected neurons, represented by systems of nonlinear ordinary stochastic differential equations—the Fitzhugh-Nagumo system with Gaussian white noise current. For a single neuron, five equations hold for the first- and second-order central moments of the voltage and recovery variables. From this system we obtain, under certain assumptions, five differential equations for the means, variances, and covariance of the two components. One may use these quantities to estimate the probability that a neuron is emitting an action potential at any given time. The differential equations are solved by numerical methods. We also perform simulations on the stochastic Fitzugh-Nagumo system and compare the results with those obtained from the differential equations for both sustained and intermittent deterministic current inputs withsuperimposed noise. For intermittent currents, which mimic synaptic input, the agreement between the analytical and simulation results for the moments is excellent. For sustained input, the analytical approximations perform well for small noise as there is excellent agreement for the moments. In addition, the probability that a neuron is spiking as obtained from the empirical distribution of the potential in the simulations gives a result almost identical to that obtained using the analytical approach. However, when there is sustained large-amplitude noise, the analytical method is only accurate for short time intervals. Using the simulation method, we study the distribution of the interspike interval directly from simulated sample paths. We confirm that noise extends the range of input currents over which (nonperiodic) spike trains may exist and investigate the dependence of such firing on the magnitude of the mean input current and the noise amplitude. For networks we find the differential equations for the means, variances, and covariances of the voltage and recovery variables and show how solving them leads to an expression for the probability that a given neuron, or given set of neurons, is firing at time t. Using such expressions one may implement dynamical rules for changing synaptic strengths directly without sampling. The present analytical method applies equally well to temporally nonhomogeneous input currents and is expected to be useful for computational studies of information processing in various nervous system centers.     

Evidence for low GluR2 AMPA receptor subunit expression at synapses in the rat basolateral amygdala

Fast excitatory synaptic responses in basolateral amygdala (BLA) neurons are mainly mediated by ionotropic glutamate receptors of the alpha-amino-3-hydroxy-5-methylisoxazole-4-propionate (AMPA) subtype. AMPA receptors containing an edited GluR2 subunit are calcium impermeable, whereas those that lack this subunit are calcium permeable and also inwardly rectifying. Here, we sought to determine the extent to which synapses in the rat BLA have AMPA receptors with GluR2 subunits. We assessed GluR2 protein expression in the BLA by immunocytochemistry with a GluR2 subunit-specific antiserum at the light and electron microscopic level; for comparison, a parallel examination was carried out in the hippocampus. We also recorded from amygdala brain slices to examine the voltage-dependent properties of AMPA receptor- mediated evoked synaptic currents in BLA principal neurons. At the light microscopic level, GluR2 immunoreactivity was localized to the perikarya and proximal dendrites of BLA neurons; dense labeling was also present over the pyramidal cell layer of hippocampal subfields CA1 and CA3. In electron micrographs from the BLA, most of the synapses were asymmetrical with pronounced postsynaptic densities (PSD). They contained clear, spherical vesicles apposed to the PSD and were predominantly onto spines (86%), indicating that they are mainly with BLA principal neurons. Only 11% of morphological synapses in the BLA were onto postsynaptic elements that showed GluR2 immunoreactivity, in contrast to hippocampal subfields CA1 and CA3 in which 76% and 71% of postsynaptic elements were labeled (p < 0.001). Synaptic staining in the BLA and hippocampus, when it occurred, was exclusively postsynaptic, and particularly heavy over the PSD. In whole-cell voltage clamp recordings, 72% of BLA principal neurons exhibited AMPA receptor-mediated synaptic currents evoked by external capsule stimulation that were inwardly rectifying. Although BLA principal neurons express perikaryal and proximal dendritic GluR2 immunoreactivity, few synapses onto these neurons express GluR2, and a preponderance of principal neurons have inwardly rectifying AMPA-mediated synaptic currents, suggesting that targeting of GluR2 to synapses is restricted. Many BLA synaptic AMPA receptors are likely to be calcium permeable and could play roles in synaptic plasticity, epileptogenesis and excitoxicity.     

Random perturbations of spiking activity in a pair of coupled neurons

We examine the effects of stochastic input currents on the firing behaviour of two coupled Type 1 or Type 2 neurons. In Hodgkin–Huxley model neurons with standard parameters, which are Type 2, in the bistable regime, synaptic transmission can initiate oscillatory joint spiking, but white noise can terminate it. In Type 1 cells (models), typified by a quadratic integrate and fire model, synaptic coupling can cause oscillatory behaviour in excitatory cells, but Gaussian white noise can again terminate it. We locally determine an approximate basin of attraction, of the periodic orbit and explain the firing behaviour in terms of the effects of noise on the probability of escape of trajectories from      

Presynaptic specificity of endocannabinoid signaling in the hippocampus

Endocannabinoids are retrograde messengers released by neurons to modulate the strength of their synaptic inputs. Endocannabinoids are thought to mediate the suppression of GABA release that follows depolarization of a hippocampal CA1 pyramidal neuron-termed "depolarization-induced suppression of inhibition" (DSI). Here, we report that DSI is absent in mice which lack cannabinoid receptor-1 (CB1). Pharmacological and kinetic evidence suggests that CB1 activation inhibits presynaptic Ca2+ channels through direct G protein inhibition. Paired recordings show that endocannabinoids selectively inhibit a subclass of synapses distinguished by their fast kinetics and large unitary conductance. Furthermore, cannabinoid-sensitive inputs are unusual among central nervous system synapses in that they use N- but not P/Q-type Ca2+ channels for neurotransmitter release. These results indicate that endocannabinoids are highly selective, rapid modulators of hippocampal inhibition.     

Muscarinic Receptor Modulation of the Cerebellar Interpositus Nucleus In Vitro

How the cerebellum carries out its functions is not clear, even for its established roles in motor control. In particular, little is known about how the cerebellar nuclei (CN) integrate their synaptic and neuromodulatory inputs to generate cerebellar output. CN neurons receive inhibitory inputs from Purkinje cells, excitatory inputs from mossy fibre and climbing fibre collaterals, as well as a variety of neuromodulatory inputs, including cholinergic inputs. In this study we tested how activation of acetylcholine receptors modulated firing rate, intrinsic properties and synaptic transmission in the CN. Using in vitro whole-cell patch clamp recordings from neurons in the interpositus nucleus, the acetylcholine receptor agonist carbachol was shown to induce a short-term increase in firing rate, increase holding current and decrease input resistance of interpositus CN neurons. Carbachol also induced long-term depression of evoked inhibitory postsynaptic currents and a short-term depression of evoked excitatory postsynaptic currents. All effects were shown to be dependent upon muscarinic acetylcholine receptor activation. Overall, the present study has identified muscarinic receptor activation as a modulator of CN activity.

     

Analysis of receptive fields revealed by in vivo patch-clamp recordings from dorsal horn neurons and in situ intracellular recordings from dorsal root ganglion neurons

It has been thought that spinal dorsal horn neurons receive convergent inputs from not only somatosensory but also visceral pathways. For instance, the referred pain is presumed to be due to the convergence of sensory inputs from cardiac and shoulder receptive fields. However, precise investigation has not been made from dorsal horn neurons yet, because of difficulty in studying the pathways from those regions by means of conventional electrophysiology. The purpose of this study is to clarify the convergent inputs to single dorsal horn neurons from wide receptive fields using an in vivo patch-clamp recording technique from the superficial spinal dorsal horn and an intracellular recording from dorsal root ganglion neurons that keep physiological connections with the peripheral sites. Identified dorsal root ganglion neurons received an input from a quite small area, about 1 x 1 mm in width of the skin. In contrast, substantia gelatinosa neurons in the spinal cord received inputs from an unexpectedly wide area of the skin. Previous extracellular recordings have, however, revealed that substantia gelatinosa neurons have small receptive field. This discrepancy is probably due mainly to an availability of the in vivo patch-clamp method to analyze sub-threshold synaptic responses. In contrast, the extracellular recording technique allows us to analyze predominantly the firing frequency of neurons. Thus, the in vivo patch-clamp recordings from dorsal horn neurons and the intracellular recordings from DRG neurons will be useful for well understanding the sensory processing in the spinal cord.     

Intracellular Neural Recording with Pure Carbon Nanotube Probes

The computational complexity of the brain depends in part on a neuron’s capacity to integrate electrochemical information from vast numbers of synaptic inputs. The measurements of synaptic activity that are crucial for mechanistic understanding of brain function are also challenging, because they require intracellular recording methods to detect and resolve millivolt- scale synaptic potentials. Although glass electrodes are widely used for intracellular recordings, novel electrodes with superior mechanical and electrical properties are desirable, because they could extend intracellular recording methods to challenging environments, including long term recordings in freely behaving animals. Carbon nanotubes (CNTs) can theoretically deliver this advance, but the difficulty of assembling CNTs has limited their application to a coating layer or assembly on a planar substrate, resulting in electrodes that are more suitable for in vivo extracellular recording or extracellular recording from isolated cells. Here we show that a novel, yet remarkably simple, millimeter-long electrode with a sub-micron tip, fabricated from self-entangled pure CNTs can be used to obtain intracellular and extracellular recordings from vertebrate neurons in vitro and in vivo. This fabrication technology provides a new method for assembling intracellular electrodes from CNTs, affording a promising opportunity to harness nanotechnology for neuroscience applications.     

Imperfect space clamp permits electrotonic interactions between inhibitory and excitatory synaptic conductances, distorting voltage clamp recordings

The voltage clamp technique is frequently used to examine the strength and composition of synaptic input to neurons. Even accounting for imperfect voltage control of the entire cell membrane ("space clamp"), it is often assumed that currents measured at the soma are a proportional indicator of the postsynaptic conductance. Here, using NEURON simulation software to model somatic recordings from morphologically realistic neurons, we show that excitatory conductances recorded in voltage clamp mode are distorted significantly by neighboring inhibitory conductances, even when the postsynaptic membrane potential starts at the reversal potential of the inhibitory conductance. Analogous effects are observed when inhibitory postsynaptic currents are recorded at the reversal potential of the excitatory conductance. Escape potentials in poorly clamped dendrites reduce the amplitude of excitatory or inhibitory postsynaptic currents recorded at the reversal potential of the other conductance. In addition, unclamped postsynaptic inhibitory conductances linearize the recorded current-voltage relationship of excitatory inputs comprising AMPAR and NMDAR-mediated components, leading to significant underestimation of the relative contribution by NMDARs, which are particularly sensitive to small perturbations in membrane potential. Voltage clamp accuracy varies substantially between neurons and dendritic arbors of different morphology; as expected, more reliable recordings are obtained from dendrites near the soma, but up to 80% of the synaptic signal on thin, distant dendrites may be lost when postsynaptic interactions are present. These limitations of the voltage clamp technique may explain how postsynaptic effects on synaptic transmission could, in some cases, be attributed incorrectly to presynaptic mechanisms.     

Modeling motor cortical operations by an attractor network of stochastic neurons

Understanding the neural computations performed by the motor cortex requires biologically plausible models that account for cell discharge patterns revealed by neurophysiological recordings. In the present study the motor cortical activity underlying movement generation is modeled as the dynamic evolution of a large fully recurrent network of stochastic spiking neurons with noise superimposed on the synaptic transmission. We show that neural representations of the learned movement trajectories can be stored in the connectivity matrix in such a way that, when activated, a particular trajectory evolves in time as a dynamic attractor of the system while individual neurons fire irregularly with large variability in their interspike intervals. Moreover, the encoding of trajectories as attractors ensures high stability of the ensemble dynamics in the presence of synaptic noise. In agreement with neurophysiological findings, the suggested model can provide a wide repertoire of specific motor behaviors, whereas the number of specialized cells and specific connections may be negligibly small if compared with the whole population engaged in trajectory retrieving. To examine the applicability of the model we study quantitatively the relationship between local geometrical and kinematic characteristics of the trajectories generated by the network. The relationship obtained as a result of simulations is close to the 2/3 power law established by psychophysical and neurophysiological studies.     

Fast inference of interactions in assemblies of stochastic integrate-and-fire neurons from spike recordings

We present two Bayesian procedures to infer the interactions and external currents in an assembly of stochastic integrate-and-fire neurons from the recording of their spiking activity. The first procedure is based on the exact calculation of the most likely time courses of the neuron membrane potentials conditioned by the recorded spikes, and is exact for a vanishing noise variance and for an instantaneous synaptic integration. The second procedure takes into account the presence of fluctuations around the most likely time courses of the potentials, and can deal with moderate noise levels. The running time of both procedures is proportional to the number S of spikes multiplied by the squared number N of neurons. The algorithms are validated on synthetic data generated by networks with known couplings and currents. We also reanalyze previously published recordings of the activity of the salamander retina (including from 32 to 40 neurons, and from 65,000 to 170,000 spikes). We study the dependence of the inferred interactions on the membrane leaking time; the differences and similarities with the classical cross-correlation analysis are discussed.     

Computation of the electroencephalogram (EEG) from network models of point neurons

The electroencephalogram (EEG) is a major tool for non-invasively studying brain function and dysfunction. Comparing experimentally recorded EEGs with neural network models is important to better interpret EEGs in terms of neural mechanisms. Most current neural network models use networks of simple point neurons. They capture important properties of cortical dynamics, and are numerically or analytically tractable. However, point neurons cannot generate an EEG, as EEG generation requires spatially separated transmembrane currents. Here, we explored how to compute an accurate approximation of a rodent’s EEG with quantities defined in point-neuron network models. We constructed different approximations (or proxies) of the EEG signal that can be computed from networks of leaky integrate-and-fire (LIF) point neurons, such as firing rates, membrane potentials, and combinations of synaptic currents. We then evaluated how well each proxy reconstructed a ground-truth EEG obtained when the synaptic currents of the LIF model network were fed into a three-dimensional network model of multicompartmental neurons with realistic morphologies. Proxies based on linear combinations of AMPA and GABA currents performed better than proxies based on firing rates or membrane potentials. A new class of proxies, based on an optimized linear combination of time-shifted AMPA and GABA currents, provided the most accurate estimate of the EEG over a wide range of network states. The new linear proxies explained 85–95% of the variance of the ground-truth EEG for a wide range of network configurations including different cell morphologies, distributions of presynaptic inputs, positions of the recording electrode, and spatial extensions of the network. Non-linear EEG proxies using a convolutional neural network (CNN) on synaptic currents increased proxy performance by a further 2–8%. Our proxies can be used to easily calculate a biologically realistic EEG signal directly from point-neuron simulations thus facilitating a quantitative comparison between computational models and experimental EEG recordings.     

Inferring synaptic inputs given a noisy voltage trace via sequential Monte Carlo methods

We discuss methods for optimally inferring the synaptic inputs to an electrotonically compact neuron, given intracellular voltage-clamp or current-clamp recordings from the postsynaptic cell. These methods are based on sequential Monte Carlo techniques ("particle filtering"). We demonstrate, on model data, that these methods can recover the time course of excitatory and inhibitory synaptic inputs accurately on a single trial. Depending on the observation noise level, no averaging over multiple trials may be required. However, excitatory inputs are consistently inferred more accurately than inhibitory inputs at physiological resting potentials, due to the stronger driving force associated with excitatory conductances. Once these synaptic input time courses are recovered, it becomes possible to fit (via tractable convex optimization techniques) models describing the relationship between the sensory stimulus and the observed synaptic input. We develop both parametric and nonparametric expectation-maximization (EM) algorithms that consist of alternating iterations between these synaptic recovery and model estimation steps. We employ a fast, robust convex optimization-based method to effectively initialize the filter; these fast methods may be of independent interest. The proposed methods could be applied to better understand the balance between excitation and inhibition in sensory processing in vivo.