Estimation of Time-Dependent Input from Neuronal Membrane Potential

An extended Hilbert transform method for reconstructing the phase from an oscillatory signal

Rhythmic activity is ubiquitous in biological systems from the cellular to organism level. Reconstructing the instantaneous phase is the first step in analyzing the essential mechanism leading to a synchronization state from the observed signals. A popular method of phase reconstruction is based on the Hilbert transform, which can only reconstruct the interpretable phase from a limited class of signals, e.g., narrow band signals. To address this issue, we propose an extended Hilbert transform method that accurately reconstructs the phase from various oscillatory signals. The proposed method is developed by analyzing the reconstruction error of the Hilbert transform method with the aid of Bedrosian's theorem. We validate the proposed method using synthetic data and show its systematically improved performance compared with the conventional Hilbert transform method with respect to accurately reconstructing the phase. Finally, we demonstrate that the proposed method is potentially useful for detecting the phase shift in an observed signal. The proposed method is expected to facilitate the study of synchronization phenomena from experimental data.

A computational approach for the inverse problem of neuronal conductances determination

The derivation by Alan Hodgkin and Andrew Huxley of their famous neuronal conductance model relied on experimental data gathered using the squid giant axon. However, the experimental determination of conductances of neurons is difficult, in particular under the presence of spatial and temporal heterogeneities, and it is also reasonable to expect variations between species or even between different types of neurons of the same species.We tackle the inverse problem of determining, given voltage data, conductances with non-uniform distribution in the simpler setting of a passive cable equation, both in a single or branched neurons. To do so, we consider the minimal error iteration, a computational technique used to solve inverse problems. We provide several numerical results showing that the method is able to provide reasonable approximations for the conductances, given enough information on the voltages, even for noisy data.

A stochastic model for interacting neurons in the olfactory bulb

We focus on interacting neurons organized in a block-layered network devoted to the information processing from the sensory system to the brain. Specifically, we consider the firing activity of olfactory sensory neurons, periglomerular, granule and mitral cells in the context of the neuronal activity of the olfactory bulb. We propose and investigate a stochastic model of a layered and modular network to describe the dynamic behavior of each prototypical neuron, taking into account both its role (excitatory/inhibitory) and its location within the network. We adopt specific Gauss-Markov processes suitable to provide reliable estimates of the firing activity of the different neurons, given their linkages. Furthermore, we study the impact of selective excitation/inhibition on the information transmission by means of simulations and numerical estimates obtained through a Volterra integral approach.

Synchrony measure for a neuron driven by excitatory and inhibitory inputs and its adaptation to experimentally-recorded data

The aim of the current work is twofold: firstly to adapt an existing method measuring the input synchrony of a neuron driven only by excitatory inputs in such a way so as to account for inhibitory inputs as well and secondly to further appropriately adapt this measure so as to be correctly utilised on experimentally-recorded data. The existing method uses the normalized pre-spike slope (NPSS) of the membrane potential, resulting from observing the slope of depolarization of the membrane potential of a neuron prior to the moment of crossing the threshold within a short period of time, to identify the response-relevant input synchrony and through it to infer the operational mode of a neuron. The first adaptation of NPSS is made such that its upper bound calculation accommodates for the higher possible slope values caused by the lower average and minimum membrane potential values due to inhibitory inputs. Results indicate that when the input spike trains arrive randomly, the modified NPSS works as expected inferring that the neuron is operating as a temporal integrator. When the input spike trains arrive in perfect synchrony though, the modified NPSS works as expected only when the level of inhibition is much higher than the level of excitation. This suggests that calculation of the upper bound of the NPSS should be a function of the ratio between excitatory and inhibitory inputs in order to be able to correctly capture perfect synchrony at a neuron's input. In addition, we effectively demonstrate a process which has to be followed when aiming to use the NPSS on real neuron recordings. This process, which relies on empirical observations of the slope of depolarisation for estimating the bounds for the range of observed interspike interval lengths, is successfully applied to experimentally-recorded data showing that through it both a real neuron's operational mode and the amount of input synchrony that caused its firing can be inferred.

Extraction of Synaptic Input Properties in Vivo

Knowledge of synaptic input is crucial for understanding synaptic integration and ultimately neural function. However, in vivo, the rates at which synaptic inputs arrive are high, so that it is typically impossible to detect single events. We show here that it is nevertheless possible to extract the properties of the events and, in particular, to extract the event rate, the synaptic time constants, and the properties of the event size distribution from in vivo voltage-clamp recordings. Applied to cerebellar interneurons, our method reveals that the synaptic input rate increases from 600 Hz during rest to 1000 Hz during locomotion, while the amplitude and shape of the synaptic events are unaffected by this state change. This method thus complements existing methods to measure neural function in vivo.

Simultaneous Bayesian Estimation of Excitatory and Inhibitory Synaptic Conductances by Exploiting Multiple Recorded Trials

Advanced statistical methods have enabled trial-by-trial inference of the underlying excitatory and inhibitory synaptic conductances (SCs) of membrane-potential recordings. Simultaneous inference of both excitatory and inhibitory SCs sheds light on the neural circuits underlying the neural activity and advances our understanding of neural information processing. Conventional Bayesian methods can infer excitatory and inhibitory SCs based on a single trial of observed membrane potential. However, if multiple recorded trials are available, this typically leads to suboptimal estimation because they neglect common statistics (of synaptic inputs (SIs)) across trials. Here, we establish a new expectation maximization (EM) algorithm that improves such single-trial Bayesian methods by exploiting multiple recorded trials to extract common SI statistics across the trials. In this paper, the proposed EM algorithm is embedded in parallel Kalman filters or particle filters for multiple recorded trials to integrate their outputs to iteratively update the common SI statistics. These statistics are then used to infer the excitatory and inhibitory SCs of individual trials. We demonstrate the superior performance of multiple-trial Kalman filtering (MtKF) and particle filtering (MtPF) relative to that of the corresponding single-trial methods. While relative estimation error of excitatory and inhibitory SCs is known to depend on the level of current injection into a cell, our numerical simulations using MtKF show that both excitatory and inhibitory SCs are reliably inferred using an optimal level of current injection. Finally, we validate the robustness and applicability of our technique through simulation studies, and we apply MtKF to in vivo data recorded from the rat barrel cortex.

Estimation of excitatory and inhibitory synaptic conductance variations in motoneurons during locomotor-like rhythmic activity

The rhythmic activity of motoneurons (MNs) that underlies locomotion in mammals is generated by synaptic inputs from the locomotor network in the spinal cord. Thus, the quantitative estimation of excitatory and inhibitory synaptic conductances is essential to understand the mechanism by which the network generates the functional motor output. Conductance estimation is obtained from the voltage-current relationship measured by voltage-clamp- or current-clamp-recording with knowledge of the leak parameters of the recorded neuron. However, it is often difficult to obtain sufficient data to estimate synaptic conductances due to technical difficulties in electrophysiological experiments using in vivo or in vitro preparations. To address this problem, we estimated the average variations in excitatory and inhibitory synaptic conductance during a locomotion cycle from a single voltage trace without measuring the leak parameters. We found that the conductance variations can be accurately reconstructed from a voltage trace of 10 cycles by analyzing synthetic data generated from a computational model. Next, the conductance variations were estimated from mouse spinal MNs in vitro during drug-induced-locomotor-like activity. We found that the peak of excitatory conductance occurred during the depolarizing phase of the locomotor cycle, whereas the peak of inhibitory conductance occurred during the hyperpolarizing phase. These results suggest that the locomotor-like activity is generated by push-pull modulation via excitatory and inhibitory synaptic inputs.

Efficient information transfer by Poisson neurons

Recently, it has been suggested that certain neurons with Poissonian spiking statistics may communicate by discontinuously switching between two levels of firing intensity. Such a situation resembles in many ways the optimal information transmission protocol for the continuous-time Poisson channel known from information theory. In this contribution we employ the classical information-theoretic results to analyze the efficiency of such a transmission from different perspectives, emphasising the neurobiological viewpoint. We address both the ultimate limits, in terms of the information capacity under metabolic cost constraints, and the achievable bounds on performance at rates below capacity with fixed decoding error probability. In doing so we discuss optimal values of experimentally measurable quantities that can be compared with the actual neuronal recordings in a future effort.

Impact of slow K(+) currents on spike generation can be described by an adaptive threshold model

A neuron that is stimulated by rectangular current injections initially responds with a high firing rate, followed by a decrease in the firing rate. This phenomenon is called spike-frequency adaptation and is usually mediated by slow K(+) currents, such as the M-type K(+) current (I M ) or the Ca(2+)-activated K(+) current (I AHP ). It is not clear how the detailed biophysical mechanisms regulate spike generation in a cortical neuron. In this study, we investigated the impact of slow K(+) currents on spike generation mechanism by reducing a detailed conductance-based neuron model. We showed that the detailed model can be reduced to a multi-timescale adaptive threshold model, and derived the formulae that describe the relationship between slow K(+) current parameters and reduced model parameters. Our analysis of the reduced model suggests that slow K(+) currents have a differential effect on the noise tolerance in neural coding.

Optimal decoding and information transmission in Hodgkin-Huxley neurons under metabolic cost constraints

Information theory quantifies the ultimate limits on reliable information transfer by means of the channel capacity. However, the channel capacity is known to be an asymptotic quantity, assuming unlimited metabolic cost and computational power. We investigate a single-compartment Hodgkin-Huxley type neuronal model under the spike-rate coding scheme and address how the metabolic cost and the decoding complexity affects the optimal information transmission. We find that the sub-threshold stimulation regime, although attaining the smallest capacity, allows for the most efficient balance between the information transmission and the metabolic cost. Furthermore, we determine post-synaptic firing rate histograms that are optimal from the information-theoretic point of view, which enables the comparison of our results with experimental data.

Estimation of the synaptic input firing rates and characterization of the stimulation effects in an auditory neuron

To understand information processing in neuronal circuits, it is important to infer how a sensory stimulus impacts on the synaptic input to a neuron. An increase in neuronal firing during the stimulation results from pure excitation or from a combination of excitation and inhibition. Here, we develop a method for estimating the rates of the excitatory and inhibitory synaptic inputs from a membrane voltage trace of a neuron. The method is based on a modified Ornstein-Uhlenbeck neuronal model, which aims to describe the stimulation effects on the synaptic input. The method is tested using a single-compartment neuron model with a realistic description of synaptic inputs, and it is applied to an intracellular voltage trace recorded from an auditory neuron in vivo. We find that the excitatory and inhibitory inputs increase during stimulation, suggesting that the acoustic stimuli are encoded by a combination of excitation and inhibition.

Can we identify non-stationary dynamics of trial-to-trial variability?

Identifying sources of the apparent variability in non-stationary scenarios is a fundamental problem in many biological data analysis settings. For instance, neurophysiological responses to the same task often vary from each repetition of the same experiment (trial) to the next. The origin and functional role of this observed variability is one of the fundamental questions in neuroscience. The nature of such trial-to-trial dynamics however remains largely elusive to current data analysis approaches. A range of strategies have been proposed in modalities such as electro-encephalography but gaining a fundamental insight into latent sources of trial-to-trial variability in neural recordings is still a major challenge. In this paper, we present a proof-of-concept study to the analysis of trial-to-trial variability dynamics founded on non-autonomous dynamical systems. At this initial stage, we evaluate the capacity of a simple statistic based on the behaviour of trajectories in classification settings, the trajectory coherence, in order to identify trial-to-trial dynamics. First, we derive the conditions leading to observable changes in datasets generated by a compact dynamical system (the Duffing equation). This canonical system plays the role of a ubiquitous model of non-stationary supervised classification problems. Second, we estimate the coherence of class-trajectories in empirically reconstructed space of system states. We show how this analysis can discern variations attributable to non-autonomous deterministic processes from stochastic fluctuations. The analyses are benchmarked using simulated and two different real datasets which have been shown to exhibit attractor dynamics. As an illustrative example, we focused on the analysis of the rat's frontal cortex ensemble dynamics during a decision-making task. Results suggest that, in line with recent hypotheses, rather than internal noise, it is the deterministic trend which most likely underlies the observed trial-to-trial variability. Thus, the empirical tool developed within this study potentially allows us to infer the source of variability in in-vivo neural recordings.

Firing clamp: a novel method for single-trial estimation of excitatory and inhibitory synaptic neuronal conductances

Understanding non-stationary neuronal activity as seen in vivo requires estimation of both excitatory and inhibitory synaptic conductances from a single trial of recording. For this purpose, we propose a new intracellular recording method, called “firing clamp.” Synaptic conductances are estimated from the characteristics of artificially evoked probe spikes, namely the spike amplitude and the mean subthreshold potential, which are sensitive to both excitatory and inhibitory synaptic input signals. The probe spikes, timed at a fixed rate, are evoked in the dynamic-clamp mode by injected meander-like current steps, with the step duration depending on neuronal membrane voltage. We test the method with perforated-patch recordings from isolated cells stimulated by external application or synaptic release of transmitter, and validate the method with simulations of a biophysically-detailed neuron model. The results are compared with the conductance estimates based on conventional current-clamp recordings.

Inferring trial-to-trial excitatory and inhibitory synaptic inputs from membrane potential using Gaussian mixture Kalman filtering

Time-varying excitatory and inhibitory synaptic inputs govern activity of neurons and process information in the brain. The importance of trial-to-trial fluctuations of synaptic inputs has recently been investigated in neuroscience. Such fluctuations are ignored in the most conventional techniques because they are removed when trials are averaged during linear regression techniques. Here, we propose a novel recursive algorithm based on Gaussian mixture Kalman filtering (GMKF) for estimating time-varying excitatory and inhibitory synaptic inputs from single trials of noisy membrane potential in current clamp recordings. The KF is followed by an expectation maximization (EM) algorithm to infer the statistical parameters (time-varying mean and variance) of the synaptic inputs in a non-parametric manner. As our proposed algorithm is repeated recursively, the inferred parameters of the mixtures are used to initiate the next iteration. Unlike other recent algorithms, our algorithm does not assume an a priori distribution from which the synaptic inputs are generated. Instead, the algorithm recursively estimates such a distribution by fitting a Gaussian mixture model (GMM). The performance of the proposed algorithms is compared to a previously proposed PF-based algorithm (Paninski et al., 2012) with several illustrative examples, assuming that the distribution of synaptic input is unknown. If noise is small, the performance of our algorithms is similar to that of the previous one. However, if noise is large, they can significantly outperform the previous proposal. These promising results suggest that our algorithm is a robust and efficient technique for estimating time varying excitatory and inhibitory synaptic conductances from single trials of membrane potential recordings.

Synaptic inhibition and excitation estimated via the time constant of membrane potential fluctuations

When recording the membrane potential, V, of a neuron it is desirable to be able to extract the synaptic input. Critically, the synaptic input is stochastic and nonreproducible so one is therefore often restricted to single-trial data. Here, we introduce means of estimating the inhibition and excitation and their confidence limits from single sweep trials. The estimates are based on the mean membrane potential, V̄, and the membrane time constant, τ. The time constant provides the total conductance ( G = capacitance/τ) and is extracted from the autocorrelation of V. The synaptic conductances can then be inferred from V̄ when approximating the neuron as a single compartment. We further employ a stochastic model to establish limits of confidence. The method is verified on models and experimental data, where the synaptic input is manipulated pharmacologically or estimated by an alternative method. The method gives best results if the synaptic input is large compared with other conductances, the intrinsic conductances have little or no time dependence or are comparably small, the ligand-gated kinetics is faster than the membrane time constant, and the majority of synaptic contacts are electrotonically close to soma (recording site). Although our data are in current clamp, the method also works in V-clamp recordings, with some minor adaptations. All custom made procedures are provided in Matlab.

Measuring input synchrony in the Ornstein-Uhlenbeck neuronal model through input parameter estimation

We present a method of estimating the input parameters and through them, the input synchrony, of a stochastic leaky integrate-and-fire neuronal model based on the Ornstein-Uhlenbeck process when it is driven by time-dependent sinusoidal input signal and noise. By driving the neuron using sinusoidal inputs, we simulate the effects of periodic synchrony on the membrane voltage and the firing of the neuron, where the peaks of the sine wave represent volleys of synchronised input spikes. Our estimation methods allow us to measure the degree of synchrony driving the neuron in terms of the input sine wave parameters, using the output spikes of the model and the membrane potential. In particular, by estimating the frequency of the synchronous input volleys and averaging the estimates of the level of input activity at corresponding intervals of the input signal, we obtain fairly accurate estimates of the baseline and peak activity of the input, which in turn define the degrees of synchrony. The same procedure is also successfully applied in estimating the baseline and peak activity of the noise. This article is part of a Special Issue entitled Neural Coding 2012.

Estimating nonstationary input signals from a single neuronal spike train

Neurons temporally integrate input signals, translating them into timed output spikes. Because neurons nonperiodically emit spikes, examining spike timing can reveal information about input signals, which are determined by activities in the populations of excitatory and inhibitory presynaptic neurons. Although a number of mathematical methods have been developed to estimate such input parameters as the mean and fluctuation of the input current, these techniques are based on the unrealistic assumption that presynaptic activity is constant over time. Here, we propose tracking temporal variations in input parameters with a two-step analysis method. First, nonstationary firing characteristics comprising the firing rate and non-Poisson irregularity are estimated from a spike train using a computationally feasible state-space algorithm. Then, information about the firing characteristics is converted into likely input parameters over time using a transformation formula, which was constructed by inverting the neuronal forward transformation of the input current to output spikes. By analyzing spike trains recorded in vivo, we found that neuronal input parameters are similar in the primary visual cortex V1 and middle temporal area, whereas parameters in the lateral geniculate nucleus of the thalamus were markedly different.

Distinguishing the Causes of Firing with the Membrane Potential Slope

In this letter, we aim to measure the relative contribution of coincidence detection and temporal integration to the firing of spikes of a simple neuron model. To this end, we develop a method to infer the degree of synchrony in an ensemble of neurons whose firing drives a single postsynaptic cell. This is accomplished by studying the effects of synchronous inputs on the membrane potential slope of the neuron and estimating the degree of response-relevant input synchrony, which determines the neuron's operational mode. The measure is calculated using the normalized slope of the membrane potential prior to the spikes fired by a neuron, and we demonstrate that it is able to distinguish between the two operational modes. By applying this measure to the membrane potential time course of a leaky integrate-and-fire neuron with the partial somatic reset mechanism, which has been shown to be the most likely candidate to reflect the mechanism used in the brain for reproducing the highly irregular firing at high rates, we show that the partial reset model operates as a temporal integrator of incoming excitatory postsynaptic potentials and that coincidence detection is not necessary for producing such high irregular firing.