Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data

On a framework of data assimilation for hyperparameter estimation of spiking neuronal networks

When handling real-world data modeled by a complex network dynamical system, the number of the parameters is often much more than the size of the data. Therefore, in many cases, it is impossible to estimate these parameters and the exact value of each parameter is frequently less interesting than the distribution of the parameters. In this paper, we aim to estimate the distribution of the parameters in the mesoscopic neuronal network model from the macroscopic experimental data, for example, the BOLD (blood oxygen level dependent) signal. Herein, we assume that the parameters of the neurons and synapses are inhomogeneous but independently and identically distributed from certain distributions with unknown hyperparameters. Thus, we estimate these hyperparameters of the distributions of the parameters, instead of estimating the parameters themselves. We formulate this problem under the framework of data assimilation and hierarchical Bayesian method and present an efficient method named Hierarchical Data Assimilation (HDA) to conduct the statistical inference on the neuronal network model with the BOLD signal data simulated by the hemodynamic model. We consider the Leaky Integral-Fire (LIF) neuronal networks with four synapses and show that the proposed algorithm can estimate the BOLD signals and the hyperparameters with high preciseness. In addition, we discuss the influence on the performance of the algorithm configuration and the LIF network model setup.

Fluctuating landscapes and heavy tails in animal behavior

Animal behavior is shaped by a myriad of mechanisms acting on a wide range of scales. This immense variability hampers quantitative reasoning and renders the identification of universal principles elusive. Through data analysis and theory, we here show that slow non-ergodic drives generally give rise to heavy-tailed statistics in behaving animals. We leverage high-resolution recordings of C. elegans locomotion to extract a self-consistent reduced order model for an inferred reaction coordinate, bridging from sub-second chaotic dynamics to long-lived stochastic transitions among metastable states. The slow mode dynamics exhibits heavy-tailed first passage time distributions and correlation functions, and we show that such heavy tails can be explained by dynamics on a time-dependent potential landscape. Inspired by these results, we introduce a generic model in which we separate faster mixing modes that evolve on a quasi-stationary potential, from slower non-ergodic modes that drive the potential landscape, and reflect slowly varying internal states. We show that, even for simple potential landscapes, heavy tails emerge when barrier heights fluctuate slowly and strongly enough. In particular, the distribution of first passage times and the correlation function can asymptote to a power law, with related exponents that depend on the strength and nature of the fluctuations. We support our theoretical findings through direct numerical simulations.

Inferring and validating mechanistic models of neural microcircuits based on spike-train data

The interpretation of neuronal spike train recordings often relies on abstract statistical models that allow for principled parameter estimation and model selection but provide only limited insights into underlying microcircuits. In contrast, mechanistic models are useful to interpret microcircuit dynamics, but are rarely quantitatively matched to experimental data due to methodological challenges. Here we present analytical methods to efficiently fit spiking circuit models to single-trial spike trains. Using derived likelihood functions, we statistically infer the mean and variance of hidden inputs, neuronal adaptation properties and connectivity for coupled integrate-and-fire neurons. Comprehensive evaluations on synthetic data, validations using ground truth in-vitro and in-vivo recordings, and comparisons with existing techniques demonstrate that parameter estimation is very accurate and efficient, even for highly subsampled networks. Our methods bridge statistical, data-driven and theoretical, model-based neurosciences at the level of spiking circuits, for the purpose of a quantitative, mechanistic interpretation of recorded neuronal population activity.

It is difficult to fit mechanistic, biophysically constrained circuit models to spike train data from in vivo extracellular recordings. Here the authors present analytical methods that enable efficient parameter estimation for integrate-and-fire circuit models and inference of the underlying connectivity structure in subsampled networks.

Reconstruction of the input signal of the leaky integrate-and-fire neuronal model from its interspike intervals

Extracting the input signal of a neuron by analyzing its spike output is an important step toward understanding how external information is coded into discrete events of action potentials and how this information is exchanged between different neurons in the nervous system. Most of the existing methods analyze this decoding problem in a stochastic framework and use probabilistic metrics such as maximum-likelihood method to determine the parameters of the input signal assuming a leaky and integrate-and-fire (LIF) model. In this article, the input signal of the LIF model is considered as a combination of orthogonal basis functions. The coefficients of the basis functions are found by minimizing the norm of the observed spikes and those generated by the estimated signal. This approach gives rise to the deterministic reconstruction of the input signal and results in a simple matrix identity through which the coefficients of the basis functions and therefore the neuronal stimulus can be identified. The inherent noise of the neuron is considered as an additional factor in the membrane potential and is treated as the disturbance in the reconstruction algorithm. The performance of the proposed scheme is evaluated by numerical simulations, and it is shown that input signals with different characteristics can be well recovered by this algorithm.

Fokker-Planck and Fortet Equation-Based Parameter Estimation for a Leaky Integrate-and-Fire Model with Sinusoidal and Stochastic Forcing

Analysis of sinusoidal noisy leaky integrate-and-fire models and comparison with experimental data are important to understand the neural code and neural synchronization and rhythms. In this paper, we propose two methods to estimate input parameters using interspike interval data only. One is based on numerical solutions of the Fokker-Planck equation, and the other is based on an integral equation, which is fulfilled by the interspike interval probability density. This generalizes previous methods tailored to stationary data to the case of time-dependent input. The main contribution is a binning method to circumvent the problems of nonstationarity, and an easy-to-implement initializer for the numerical procedures. The methods are compared on simulated data.

LIST OF ABBREVIATIONS LIF

Leaky integrate-and-fireISI: Interspike intervalSDE: Stochastic differential equationPDE: Partial differential equation.

A combined method to estimate parameters of the thalamocortical model from a heavily noise-corrupted time series of action potential

A combined method composing of the unscented Kalman filter (UKF) and the synchronization-based method is proposed for estimating electrophysiological variables and parameters of a thalamocortical (TC) neuron model, which is commonly used for studying Parkinson's disease for its relay role of connecting the basal ganglia and the cortex. In this work, we take into account the condition when only the time series of action potential with heavy noise are available. Numerical results demonstrate that not only this method can estimate model parameters from the extracted time series of action potential successfully but also the effect of its estimation is much better than the only use of the UKF or synchronization-based method, with a higher accuracy and a better robustness against noise, especially under the severe noise conditions. Considering the rather important role of TC neuron in the normal and pathological brain functions, the exploration of the method to estimate the critical parameters could have important implications for the study of its nonlinear dynamics and further treatment of Parkinson's disease.

Estimating nonstationary inputs from a single spike train based on a neuron model with adaptation

Because every spike of a neuron is determined by input signals, a train of spikes may contain information about the dynamics of unobserved neurons. A state-space method based on the leaky integrate-and-fire model, describing neuronal transformation from input signals to a spike train has been proposed for tracking input parameters represented by their mean and fluctuation [11]. In the present paper, we propose to make the estimation more realistic by adopting an LIF model augmented with an adaptive moving threshold. Moreover, because the direct state-space method is computationally infeasible for a data set comprising thousands of spikes, we further develop a practical method for transforming instantaneous firing characteristics back to input parameters. The instantaneous firing characteristics, represented by the firing rate and non-Poisson irregularity, can be estimated using a computationally feasible algorithm. We applied our proposed methods to synthetic data to clarify that they perform well.

A unified approach to linking experimental, statistical and computational analysis of spike train data

A fundamental issue in neuroscience is how to identify the multiple biophysical mechanisms through which neurons generate observed patterns of spiking activity. In previous work, we proposed a method for linking observed patterns of spiking activity to specific biophysical mechanisms based on a state space modeling framework and a sequential Monte Carlo, or particle filter, estimation algorithm. We have shown, in simulation, that this approach is able to identify a space of simple biophysical models that were consistent with observed spiking data (and included the model that generated the data), but have yet to demonstrate the application of the method to identify realistic currents from real spike train data. Here, we apply the particle filter to spiking data recorded from rat layer V cortical neurons, and correctly identify the dynamics of an slow, intrinsic current. The underlying intrinsic current is successfully identified in four distinct neurons, even though the cells exhibit two distinct classes of spiking activity: regular spiking and bursting. This approach – linking statistical, computational, and experimental neuroscience – provides an effective technique to constrain detailed biophysical models to specific mechanisms consistent with observed spike train data.

Estimating inputs and an internal neuronal parameter from a single spike train

Because neurons are integrating input signals and translating them into timed output spikes, examining spike timing may reveal information about inputs, such as population activities of excitatory and inhibitory presynaptic neurons. Here we construct a state-space method for estimating not only such extrinsic parameters, but also an intrinsic neuronal parameter such as the membrane time constant from a single spike train.

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.

A sequential Monte Carlo approach to estimate biophysical neural models from spikes

Realistic computational models of neuronal activity typically involve many variables and parameters, most of which remain unknown or poorly constrained. Moreover, experimental observations of the neuronal system are typically limited to the times of action potentials, or spikes. One important component of developing a computational model is the optimal incorporation of these sparse experimental data. Here, we use point process statistical theory to develop a procedure for estimating parameters and hidden variables in neuronal computational models given only the observed spike times. We discuss the implementation of a sequential Monte Carlo method for this procedure and apply it to three simulated examples of neuronal spiking activity. We also address the issues of model identification and misspecification, and show that accurate estimates of model parameters and hidden variables are possible given only spike time data.

Spiking Neurons and the First Passage Problem

We derive a model of a neuron’s interspike interval probability density through analysis of the first passage problem. The fit of our expression to retinal ganglion cell laboratory data extracts three physiologically relevant parameters, with which our model yields input-output features that conform to laboratory results. Preliminary analysis suggests that under common circumstances, local circuitry readjusts these parameters with changes in firing rate and so endeavors to faithfully replicate an input signal. Further results suggest that the so-called principle of sloppy workmanship also plays a role in evolution’s choice of these parameters.

Motoneuron membrane potentials follow a time inhomogeneous jump diffusion process

Stochastic leaky integrate-and-fire models are popular due to their simplicity and statistical tractability. They have been widely applied to gain understanding of the underlying mechanisms for spike timing in neurons, and have served as building blocks for more elaborate models. Especially the Ornstein-Uhlenbeck process is popular to describe the stochastic fluctuations in the membrane potential of a neuron, but also other models like the square-root model or models with a non-linear drift are sometimes applied. Data that can be described by such models have to be stationary and thus, the simple models can only be applied over short time windows. However, experimental data show varying time constants, state dependent noise, a graded firing threshold and time-inhomogeneous input. In the present study we build a jump diffusion model that incorporates these features, and introduce a firing mechanism with a state dependent intensity. In addition, we suggest statistical methods to estimate all unknown quantities and apply these to analyze turtle motoneuron membrane potentials. Finally, simulated and real data are compared and discussed. We find that a square-root diffusion describes the data much better than an Ornstein-Uhlenbeck process with constant diffusion coefficient. Further, the membrane time constant decreases with increasing depolarization, as expected from the increase in synaptic conductance. The network activity, which the neuron is exposed to, can be reasonably estimated to be a threshold version of the nerve output from the network. Moreover, the spiking characteristics are well described by a Poisson spike train with an intensity depending exponentially on the membrane potential.

Statistical technique for analysing functional connectivity of multiple spike trains

A new statistical technique, the Cox method, used for analysing functional connectivity of simultaneously recorded multiple spike trains is presented. This method is based on the theory of modulated renewal processes and it estimates a vector of influence strengths from multiple spike trains (called reference trains) to the selected (target) spike train. Selecting another target spike train and repeating the calculation of the influence strengths from the reference spike trains enables researchers to find all functional connections among multiple spike trains. In order to study functional connectivity an "influence function" is identified. This function recognises the specificity of neuronal interactions and reflects the dynamics of postsynaptic potential. In comparison to existing techniques, the Cox method has the following advantages: it does not use bins (binless method); it is applicable to cases where the sample size is small; it is sufficiently sensitive such that it estimates weak influences; it supports the simultaneous analysis of multiple influences; it is able to identify a correct connectivity scheme in difficult cases of "common source" or "indirect" connectivity. The Cox method has been thoroughly tested using multiple sets of data generated by the neural network model of the leaky integrate and fire neurons with a prescribed architecture of connections. The results suggest that this method is highly successful for analysing functional connectivity of simultaneously recorded multiple spike trains.

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.

Estimating input parameters from intracellular recordings in the Feller neuronal model

We study the estimation of the input parameters in a Feller neuronal model from a trajectory of the membrane potential sampled at discrete times. These input parameters are identified with the drift and the infinitesimal variance of the underlying stochastic diffusion process with multiplicative noise. The state space of the process is restricted from below by an inaccessible boundary. Further, the model is characterized by the presence of an absorbing threshold, the first hitting of which determines the length of each trajectory and which constrains the state space from above. We compare, both in the presence and in the absence of the absorbing threshold, the efficiency of different known estimators. In addition, we propose an estimator for the drift term, which is proved to be more efficient than the others, at least in the explored range of the parameters. The presence of the threshold makes the estimates of the drift term biased, and two methods to correct it are proposed.

State-space algorithms for estimating spike rate functions

The accurate characterization of spike firing rates including the determination of when changes in activity occur is a fundamental issue in the analysis of neurophysiological data. Here we describe a state-space model for estimating the spike rate function that provides a maximum likelihood estimate of the spike rate, model goodness-of-fit assessments, as well as confidence intervals for the spike rate function and any other associated quantities of interest. Using simulated spike data, we first compare the performance of the state-space approach with that of Bayesian adaptive regression splines (BARS) and a simple cubic spline smoothing algorithm. We show that the state-space model is computationally efficient and comparable with other spline approaches. Our results suggest both a theoretically sound and practical approach for estimating spike rate functions that is applicable to a wide range of neurophysiological data.

Effect of stimulation on the input parameters of stochastic leaky integrate-and-fire neuronal model

The Ornstein-Uhlenbeck neuronal model is specified by two types of parameters. One type corresponds to the properties of the neuronal membrane, whereas the second type (local average rate of the membrane depolarization and its variability) corresponds to the input of the neuron. In this article, we estimate the parameters of the second type from an intracellular record during neuronal firing caused by stimulation (audio signal). We compare the obtained estimates with those from the spontaneous part of the record. As predicted from the model construction, the values of the input parameters are larger for the periods when neuron is stimulated than for the spontaneous ones. Finally, the firing regimen of the model is checked. It is confirmed that the neuron is in the suprathreshold regimen during the stimulation.

Are the input parameters of white noise driven integrate and fire neurons uniquely determined by rate and CV?

Integrate and fire (IF) neurons have found widespread applications in computational neuroscience. Particularly important are stochastic versions of these models where the driving consists of a synaptic input modeled as white Gaussian noise with mean mu and noise intensity D. Different IF models have been proposed, the firing statistics of which depends nontrivially on the input parameters mu and D. In order to compare these models among each other, one must first specify the correspondence between their parameters. This can be done by determining which set of parameters (mu,D) of each model is associated with a given set of basic firing statistics as, for instance, the firing rate and the coefficient of variation (CV) of the interspike interval (ISI). However, it is not clear a priori whether for a given firing rate and CV there is only one unique choice of input parameters for each model. Here we review the dependence of rate and CV on input parameters for the perfect, leaky, and quadratic IF neuron models and show analytically that indeed in these three models the firing rate and the CV uniquely determine the input parameters.

The quantitative single-neuron modeling competition

As large-scale, detailed network modeling projects are flourishing in the field of computational neuroscience, it is more and more important to design single neuron models that not only capture qualitative features of real neurons but are quantitatively accurate in silico representations of those. Recent years have seen substantial effort being put in the development of algorithms for the systematic evaluation and optimization of neuron models with respect to electrophysiological data. It is however difficult to compare these methods because of the lack of appropriate benchmark tests. Here, we describe one such effort of providing the community with a standardized set of tests to quantify the performances of single neuron models. Our effort takes the form of a yearly challenge similar to the ones which have been present in the machine learning community for some time. This paper gives an account of the first two challenges which took place in 2007 and 2008 and discusses future directions. The results of the competition suggest that best performance on data obtained from single or double electrode current or conductance injection is achieved by models that combine features of standard leaky integrate-and-fire models with a second variable reflecting adaptation, refractoriness, or a dynamic threshold.

A review of the methods for signal estimation in stochastic diffusion leaky integrate-and-fire neuronal models

Parameters in diffusion neuronal models are divided into two groups; intrinsic and input parameters. Intrinsic parameters are related to the properties of the neuronal membrane and are assumed to be known throughout the paper. Input parameters characterize processes generated outside the neuron and methods for their estimation are reviewed here. Two examples of the diffusion neuronal model, which are based on the integrate-and-fire concept, are investigated--the Ornstein--Uhlenbeck model as the most common one and the Feller model as an illustration of state-dependent behavior in modeling the neuronal input. Two types of experimental data are assumed-intracellular describing the membrane trajectories and extracellular resulting in knowledge of the interspike intervals. The literature on estimation from the trajectories of the diffusion process is extensive and thus the stress in this review is set on the inference made from the interspike intervals.