Approximate Methods for State-Space Models

Marked point process variational autoencoder with applications to unsorted spiking activities

Spike train modeling across large neural populations is a powerful tool for understanding how neurons code information in a coordinated manner. Recent studies have employed marked point processes in neural population modeling. The marked point process is a stochastic process that generates a sequence of events with marks. Spike train models based on such processes use the waveform features of spikes as marks and express the generative structure of the unsorted spikes without applying spike sorting. In such modeling, the goal is to estimate the joint mark intensity that describes how observed covariates or hidden states (e.g., animal behaviors, animal internal states, and experimental conditions) influence unsorted spikes. A major issue with this approach is that existing joint mark intensity models are not designed to capture high-dimensional and highly nonlinear observations. To address this limitation, we propose a new joint mark intensity model based on a variational autoencoder, capable of representing the dependency structure of unsorted spikes on observed covariates or hidden states in a data-driven manner. Our model defines the joint mark intensity as a latent variable model, where a neural network decoder transforms a shared latent variable into states and marks. With our model, we derive a new log-likelihood lower bound by exploiting the variational evidence lower bound and upper bound (e.g., the χ upper bound) and use this new lower bound for parameter estimation. To demonstrate the strength of this approach, we integrate our model into a state space model with a nonlinear embedding to capture the hidden state dynamics underlying the observed covariates and unsorted spikes. This enables us to reconstruct covariates from unsorted spikes, known as neural decoding. Our model achieves superior performance in prediction and decoding tasks for synthetic data and the spiking activities of place cells.

Extended Variational Message Passing for Automated Approximate Bayesian Inference

Variational Message Passing (VMP) provides an automatable and efficient algorithmic framework for approximating Bayesian inference in factorized probabilistic models that consist of conjugate exponential family distributions. The automation of Bayesian inference tasks is very important since many data processing problems can be formulated as inference tasks on a generative probabilistic model. However, accurate generative models may also contain deterministic and possibly nonlinear variable mappings and non-conjugate factor pairs that complicate the automatic execution of the VMP algorithm. In this paper, we show that executing VMP in complex models relies on the ability to compute the expectations of the statistics of hidden variables. We extend the applicability of VMP by approximating the required expectation quantities in appropriate cases by importance sampling and Laplace approximation. As a result, the proposed Extended VMP (EVMP) approach supports automated efficient inference for a very wide range of probabilistic model specifications. We implemented EVMP in the Julia language in the probabilistic programming package ForneyLab.jl and show by a number of examples that EVMP renders an almost universal inference engine for factorized probabilistic models.

The Discriminative Kalman Filter for Bayesian Filtering with Nonlinear and Nongaussian Observation Models

The Kalman filter provides a simple and efficient algorithm to compute the posterior distribution for state-space models where both the latent state and measurement models are linear and gaussian. Extensions to the Kalman filter, including the extended and unscented Kalman filters, incorporate linearizations for models where the observation model p ( observation | state ) is nonlinear. We argue that in many cases, a model for p ( state | observation ) proves both easier to learn and more accurate for latent state estimation. Approximating p ( state | observation ) as gaussian leads to a new filtering algorithm, the discriminative Kalman filter (DKF), which can perform well even when p ( observation | state ) is highly nonlinear and/or nongaussian. The approximation, motivated by the Bernstein-von Mises theorem, improves as the dimensionality of the observations increases. The DKF has computational complexity similar to the Kalman filter, allowing it in some cases to perform much faster than particle filters with similar precision, while better accounting for nonlinear and nongaussian observation models than Kalman-based extensions. When the observation model must be learned from training data prior to filtering, off-the-shelf nonlinear and nonparametric regression techniques can provide a gaussian model for p ( observation | state ) that cleanly integrates with the DKF. As part of the BrainGate2 clinical trial, we successfully implemented gaussian process regression with the DKF framework in a brain-computer interface to provide real-time, closed-loop cursor control to a person with a complete spinal cord injury. In this letter, we explore the theory underlying the DKF, exhibit some illustrative examples, and outline potential extensions.

Reconstructing neuronal circuitry from parallel spike trains

State-of-the-art techniques allow researchers to record large numbers of spike trains in parallel for many hours. With enough such data, we should be able to infer the connectivity among neurons. Here we develop a method for reconstructing neuronal circuitry by applying a generalized linear model (GLM) to spike cross-correlations. Our method estimates connections between neurons in units of postsynaptic potentials and the amount of spike recordings needed to verify connections. The performance of inference is optimized by counting the estimation errors using synthetic data. This method is superior to other established methods in correctly estimating connectivity. By applying our method to rat hippocampal data, we show that the types of estimated connections match the results inferred from other physiological cues. Thus our method provides the means to build a circuit diagram from recorded spike trains, thereby providing a basis for elucidating the differences in information processing in different brain regions.

Current techniques have enabled the simultaneous collection of spike train data from large numbers of neurons. Here, the authors report a method to infer the underlying neural circuit connectivity diagram based on a generalized linear model applied to spike cross-correlations between neurons.

Identifying nonlinear dynamical systems via generative recurrent neural networks with applications to fMRI

A major tenet in theoretical neuroscience is that cognitive and behavioral processes are ultimately implemented in terms of the neural system dynamics. Accordingly, a major aim for the analysis of neurophysiological measurements should lie in the identification of the computational dynamics underlying task processing. Here we advance a state space model (SSM) based on generative piecewise-linear recurrent neural networks (PLRNN) to assess dynamics from neuroimaging data. In contrast to many other nonlinear time series models which have been proposed for reconstructing latent dynamics, our model is easily interpretable in neural terms, amenable to systematic dynamical systems analysis of the resulting set of equations, and can straightforwardly be transformed into an equivalent continuous-time dynamical system. The major contributions of this paper are the introduction of a new observation model suitable for functional magnetic resonance imaging (fMRI) coupled to the latent PLRNN, an efficient stepwise training procedure that forces the latent model to capture the 'true' underlying dynamics rather than just fitting (or predicting) the observations, and of an empirical measure based on the Kullback-Leibler divergence to evaluate from empirical time series how well this goal of approximating the underlying dynamics has been achieved. We validate and illustrate the power of our approach on simulated 'ground-truth' dynamical systems as well as on experimental fMRI time series, and demonstrate that the learnt dynamics harbors task-related nonlinear structure that a linear dynamical model fails to capture. Given that fMRI is one of the most common techniques for measuring brain activity non-invasively in human subjects, this approach may provide a novel step toward analyzing aberrant (nonlinear) dynamics for clinical assessment or neuroscientific research.

Decoding of Neural Data Using Cohomological Feature Extraction

We introduce a novel data-driven approach to discover and decode features in the neural code coming from large population neural recordings with minimal assumptions, using cohomological feature extraction. We apply our approach to neural recordings of mice moving freely in a box, where we find a circular feature. We then observe that the decoded value corresponds well to the head direction of the mouse. Thus, we capture head direction cells and decode the head direction from the neural population activity without having to process the mouse's behavior. Interestingly, the decoded values convey more information about the neural activity than the tracked head direction does, with differences that have some spatial organization. Finally, we note that the residual population activity, after the head direction has been accounted for, retains some low-dimensional structure that is correlated with the speed of the mouse.

Data-Driven Transducer Design and Identification for Internally-Paced Motor Brain Computer Interfaces: A Review

Brain-Computer Interfaces (BCIs) are systems that establish a direct communication pathway between the users' brain activity and external effectors. They offer the potential to improve the quality of life of motor-impaired patients. Motor BCIs aim to permit severely motor-impaired users to regain limb mobility by controlling orthoses or prostheses. In particular, motor BCI systems benefit patients if the decoded actions reflect the users' intentions with an accuracy that enables them to efficiently interact with their environment. One of the main challenges of BCI systems is to adapt the BCI's signal translation blocks to the user to reach a high decoding accuracy. This paper will review the literature of data-driven and user-specific transducer design and identification approaches and it focuses on internally-paced motor BCIs. In particular, continuous kinematic biomimetic and mental-task decoders are reviewed. Furthermore, static and dynamic decoding approaches, linear and non-linear decoding, offline and real-time identification algorithms are considered. The current progress and challenges related to the design of clinical-compatible motor BCI transducers are additionally discussed.

Modeling, Detecting, and Tracking Freezing of Gait in Parkinson Disease Using Inertial Sensors

In this paper, we develop new methods to automatically detect the onset and duration of freezing of gait (FOG) in people with Parkinson disease (PD) in real time, using inertial sensors. We first build a physical model that describes the trembling motion during the FOG events. Then, we design a generalized likelihood ratio test framework to develop a two-stage detector for determining the zero-velocity and trembling events during gait. Thereafter, to filter out falsely detected FOG events, we develop a point-process filter that combines the output of the detectors with information about the speed of the foot, provided by a foot-mounted inertial navigation system. We computed the probability of FOG by using the point-process filter to determine the onset and duration of the FOG event. Finally, we validate the performance of the proposed system design using real data obtained from people with PD who performed a set of gait tasks. We compare our FOG detection results with an existing method that only uses accelerometer data. The results indicate that our method yields 81.03% accuracy in detecting FOG events and a threefold decrease in the false-alarm rate relative to the existing method.

Firing rate estimation using infinite mixture models and its application to neural decoding

Neural decoding is a framework for reconstructing external stimuli from spike trains recorded by various neural recordings. Kloosterman et al. proposed a new decoding method using marked point processes (Kloosterman F, Layton SP, Chen Z, Wilson MA. J Neurophysiol 111: 217-227, 2014). This method does not require spike sorting and thereby improves decoding accuracy dramatically. In this method, they used kernel density estimation to estimate intensity functions of marked point processes. However, the use of kernel density estimation causes problems such as low decoding accuracy and high computational costs. To overcome these problems, we propose a new decoding method using infinite mixture models to estimate intensity. The proposed method improves decoding performance in terms of accuracy and computational speed. We apply the proposed method to simulation and experimental data to verify its performance.NEW & NOTEWORTHY We propose a new neural decoding method using infinite mixture models and nonparametric Bayesian statistics. The proposed method improves decoding performance in terms of accuracy and computation speed. We have successfully applied the proposed method to position decoding from spike trains recorded in a rat hippocampus.

A state space approach for piecewise-linear recurrent neural networks for identifying computational dynamics from neural measurements

The computational and cognitive properties of neural systems are often thought to be implemented in terms of their (stochastic) network dynamics. Hence, recovering the system dynamics from experimentally observed neuronal time series, like multiple single-unit recordings or neuroimaging data, is an important step toward understanding its computations. Ideally, one would not only seek a (lower-dimensional) state space representation of the dynamics, but would wish to have access to its statistical properties and their generative equations for in-depth analysis. Recurrent neural networks (RNNs) are a computationally powerful and dynamically universal formal framework which has been extensively studied from both the computational and the dynamical systems perspective. Here we develop a semi-analytical maximum-likelihood estimation scheme for piecewise-linear RNNs (PLRNNs) within the statistical framework of state space models, which accounts for noise in both the underlying latent dynamics and the observation process. The Expectation-Maximization algorithm is used to infer the latent state distribution, through a global Laplace approximation, and the PLRNN parameters iteratively. After validating the procedure on toy examples, and using inference through particle filters for comparison, the approach is applied to multiple single-unit recordings from the rodent anterior cingulate cortex (ACC) obtained during performance of a classical working memory task, delayed alternation. Models estimated from kernel-smoothed spike time data were able to capture the essential computational dynamics underlying task performance, including stimulus-selective delay activity. The estimated models were rarely multi-stable, however, but rather were tuned to exhibit slow dynamics in the vicinity of a bifurcation point. In summary, the present work advances a semi-analytical (thus reasonably fast) maximum-likelihood estimation framework for PLRNNs that may enable to recover relevant aspects of the nonlinear dynamics underlying observed neuronal time series, and directly link these to computational properties.

Variational Latent Gaussian Process for Recovering Single-Trial Dynamics from Population Spike Trains

When governed by underlying low-dimensional dynamics, the interdependence of simultaneously recorded populations of neurons can be explained by a small number of shared factors, or a low-dimensional trajectory. Recovering these latent trajectories, particularly from single-trial population recordings, may help us understand the dynamics that drive neural computation. However, due to the biophysical constraints and noise in the spike trains, inferring trajectories from data is a challenging statistical problem in general. Here, we propose a practical and efficient inference method, the variational latent gaussian process (vLGP). The vLGP combines a generative model with a history-dependent point process observation, together with a smoothness prior on the latent trajectories. The vLGP improves on earlier methods for recovering latent trajectories, which assume either observation models inappropriate for point processes or linear dynamics. We compare and validate vLGP on both simulated data sets and population recordings from the primary visual cortex. In the V1 data set, we find that vLGP achieves substantially higher performance than previous methods for predicting omitted spike trains, as well as capturing both the toroidal topology of visual stimuli space and the noise correlation. These results show that vLGP is a robust method with the potential to reveal hidden neural dynamics from large-scale neural recordings.

Recasting brain-machine interface design from a physical control system perspective

With the goal of improving the quality of life for people suffering from various motor control disorders, brain-machine interfaces provide direct neural control of prosthetic devices by translating neural signals into control signals. These systems act by reading motor intent signals directly from the brain and using them to control, for example, the movement of a cursor on a computer screen. Over the past two decades, much attention has been devoted to the decoding problem: how should recorded neural activity be translated into the movement of the cursor? Most approaches have focused on this problem from an estimation standpoint, i.e., decoders are designed to return the best estimate of motor intent possible, under various sets of assumptions about how the recorded neural signals represent motor intent. Here we recast the decoder design problem from a physical control system perspective, and investigate how various classes of decoders lead to different types of physical systems for the subject to control. This framework leads to new interpretations of why certain types of decoders have been shown to perform better than others. These results have implications for understanding how motor neurons are recruited to perform various tasks, and may lend insight into the brain's ability to conceptualize artificial systems.

Single-unit activity, threshold crossings, and local field potentials in motor cortex differentially encode reach kinematics

A diversity of signals can be recorded with extracellular electrodes. It remains unclear whether different signal types convey similar or different information and whether they capture the same or different underlying neural phenomena. Some researchers focus on spiking activity, while others examine local field potentials, and still others posit that these are fundamentally the same signals. We examined the similarities and differences in the information contained in four signal types recorded simultaneously from multielectrode arrays implanted in primary motor cortex: well-isolated action potentials from putative single units, multiunit threshold crossings, and local field potentials (LFPs) at two distinct frequency bands. We quantified the tuning of these signal types to kinematic parameters of reaching movements. We found 1) threshold crossing activity is not a proxy for single-unit activity; 2) when examined on individual electrodes, threshold crossing activity more closely resembles LFP activity at frequencies between 100 and 300 Hz than it does single-unit activity; 3) when examined across multiple electrodes, threshold crossing activity and LFP integrate neural activity at different spatial scales; and 4) LFP power in the "beta band" (between 10 and 40 Hz) is a reliable indicator of movement onset but does not encode kinematic features on an instant-by-instant basis. These results show that the diverse signals recorded from extracellular electrodes provide somewhat distinct and complementary information. It may be that these signal types arise from biological phenomena that are partially distinct. These results also have practical implications for harnessing richer signals to improve brain-machine interface control.

To sort or not to sort: the impact of spike-sorting on neural decoding performance

OBJECTIVE

Brain-computer interfaces (BCIs) are a promising technology for restoring motor ability to paralyzed patients. Spiking-based BCIs have successfully been used in clinical trials to control multi-degree-of-freedom robotic devices. Current implementations of these devices require a lengthy spike-sorting step, which is an obstacle to moving this technology from the lab to the clinic. A viable alternative is to avoid spike-sorting, treating all threshold crossings of the voltage waveform on an electrode as coming from one putative neuron. It is not known, however, how much decoding information might be lost by ignoring spike identity.

APPROACH

We present a full analysis of the effects of spike-sorting schemes on decoding performance. Specifically, we compare how well two common decoders, the optimal linear estimator and the Kalman filter, reconstruct the arm movements of non-human primates performing reaching tasks, when receiving input from various sorting schemes. The schemes we tested included: using threshold crossings without spike-sorting; expert-sorting discarding the noise; expert-sorting, including the noise as if it were another neuron; and automatic spike-sorting using waveform features. We also decoded from a joint statistical model for the waveforms and tuning curves, which does not involve an explicit spike-sorting step.

MAIN RESULTS

Discarding the threshold crossings that cannot be assigned to neurons degrades decoding: no spikes should be discarded. Decoding based on spike-sorted units outperforms decoding based on electrodes voltage crossings: spike-sorting is useful. The four waveform based spike-sorting methods tested here yield similar decoding efficiencies: a fast and simple method is competitive. Decoding using the joint waveform and tuning model shows promise but is not consistently superior.

SIGNIFICANCE

Our results indicate that simple automated spike-sorting performs as well as the more computationally or manually intensive methods used here. Even basic spike-sorting adds value to the low-threshold waveform-crossing methods often employed in BCI decoding.

Motor cortical correlates of arm resting in the context of a reaching task and implications for prosthetic control

Prosthetic devices are being developed to restore movement for motor-impaired individuals. A robotic arm can be controlled based on models that relate motor-cortical ensemble activity to kinematic parameters. The models are typically built and validated on data from structured trial periods during which a subject actively performs specific movements, but real-world prosthetic devices will need to operate correctly during rest periods as well. To develop a model of motor cortical modulation during rest, we trained monkeys (Macaca mulatta) to perform a reaching task with their own arm while recording motor-cortical single-unit activity. When a monkey spontaneously put its arm down to rest between trials, our traditional movement decoder produced a nonzero velocity prediction, which would cause undesired motion when applied to a prosthetic arm. During these rest periods, a marked shift was found in individual units' tuning functions. The activity pattern of the whole population during rest (Idle state) was highly distinct from that during reaching movements (Active state), allowing us to predict arm resting from instantaneous firing rates with 98% accuracy using a simple classifier. By cascading this state classifier and the movement decoder, we were able to predict zero velocity correctly, which would avoid undesired motion in a prosthetic application. Interestingly, firing rates during hold periods followed the Active pattern even though hold kinematics were similar to those during rest with near-zero velocity. These findings expand our concept of motor-cortical function by showing that population activity reflects behavioral context in addition to the direct parameters of the movement itself.

An overview of Bayesian methods for neural spike train analysis

Neural spike train analysis is an important task in computational neuroscience which aims to understand neural mechanisms and gain insights into neural circuits. With the advancement of multielectrode recording and imaging technologies, it has become increasingly demanding to develop statistical tools for analyzing large neuronal ensemble spike activity. Here we present a tutorial overview of Bayesian methods and their representative applications in neural spike train analysis, at both single neuron and population levels. On the theoretical side, we focus on various approximate Bayesian inference techniques as applied to latent state and parameter estimation. On the application side, the topics include spike sorting, tuning curve estimation, neural encoding and decoding, deconvolution of spike trains from calcium imaging signals, and inference of neuronal functional connectivity and synchrony. Some research challenges and opportunities for neural spike train analysis are discussed.

Information transmission using non-poisson regular firing

In many cortical areas, neural spike trains do not follow a Poisson process. In this study, we investigate a possible benefit of non-Poisson spiking for information transmission by studying the minimal rate fluctuation that can be detected by a Bayesian estimator. The idea is that an inhomogeneous Poisson process may make it difficult for downstream decoders to resolve subtle changes in rate fluctuation, but by using a more regular non-Poisson process, the nervous system can make rate fluctuations easier to detect. We evaluate the degree to which regular firing reduces the rate fluctuation detection threshold. We find that the threshold for detection is reduced in proportion to the coefficient of variation of interspike intervals.

Coupling Time Decoding and Trajectory Decoding using a Target-Included Model in the Motor Cortex

Significant progress has been made within the last decade in motor cortical decoding that predicts movement behaviors from population neuronal activity in the motor cortex. A majority of these decoding methods have focused on estimating a subject's hand trajectory in a continuous movement. We recently proposed a time identification decoding approach and showed that if a stereotyped movement is well represented by a sequence of targets (or landmarks), then the main structure of the movement can be reconstructed by detecting the reaching times at those targets. Both trajectory decoding and landmark-time decoding have their particular advantages, whereas a coupling of these two different strategies has not been examined. In this article we propose a synergy that comes from combining these two approaches for a stereotyped movement under a linear state-space framework. We develop a new decoding procedure based on a forward-backward propagation where the target is used in the initial stage in the backward step. Experimental results show that the new method significantly improves decoding accuracy over the non-target-included models. Furthermore, the coupling based on the new target-included method effectively combines the time decoding and trajectory decoding and further improves the decoding accuracy.

Markov chain Monte Carlo methods for state-space models with point process observations

This letter considers how a number of modern Markov chain Monte Carlo (MCMC) methods can be applied for parameter estimation and inference in state-space models with point process observations. We quantified the efficiencies of these MCMC methods on synthetic data, and our results suggest that the Reimannian manifold Hamiltonian Monte Carlo method offers the best performance. We further compared such a method with a previously tested variational Bayes method on two experimental data sets. Results indicate similar performance on the large data sets and superior performance on small ones. The work offers an extensive suite of MCMC algorithms evaluated on an important class of models for physiological signal analysis.

A unified point process probabilistic framework to assess heartbeat dynamics and autonomic cardiovascular control

In recent years, time-varying inhomogeneous point process models have been introduced for assessment of instantaneous heartbeat dynamics as well as specific cardiovascular control mechanisms and hemodynamics. Assessment of the model's statistics is established through the Wiener-Volterra theory and a multivariate autoregressive (AR) structure. A variety of instantaneous cardiovascular metrics, such as heart rate (HR), heart rate variability (HRV), respiratory sinus arrhythmia (RSA), and baroreceptor-cardiac reflex (baroreflex) sensitivity (BRS), are derived within a parametric framework and instantaneously updated with adaptive and local maximum likelihood estimation algorithms. Inclusion of second-order non-linearities, with subsequent bispectral quantification in the frequency domain, further allows for definition of instantaneous metrics of non-linearity. We here present a comprehensive review of the devised methods as applied to experimental recordings from healthy subjects during propofol anesthesia. Collective results reveal interesting dynamic trends across the different pharmacological interventions operated within each anesthesia session, confirming the ability of the algorithm to track important changes in cardiorespiratory elicited interactions, and pointing at our mathematical approach as a promising monitoring tool for an accurate, non-invasive assessment in clinical practice. We also discuss the limitations and other alternative modeling strategies of our point process approach.

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.

Comparison of brain-computer interface decoding algorithms in open-loop and closed-loop control

Neuroprosthetic devices such as a computer cursor can be controlled by the activity of cortical neurons when an appropriate algorithm is used to decode motor intention. Algorithms which have been proposed for this purpose range from the simple population vector algorithm (PVA) and optimal linear estimator (OLE) to various versions of Bayesian decoders. Although Bayesian decoders typically provide the most accurate off-line reconstructions, it is not known which model assumptions in these algorithms are critical for improving decoding performance. Furthermore, it is not necessarily true that improvements (or deficits) in off-line reconstruction will translate into improvements (or deficits) in on-line control, as the subject might compensate for the specifics of the decoder in use at the time. Here we show that by comparing the performance of nine decoders, assumptions about uniformly distributed preferred directions and the way the cursor trajectories are smoothed have the most impact on decoder performance in off-line reconstruction, while assumptions about tuning curve linearity and spike count variance play relatively minor roles. In on-line control, subjects compensate for directional biases caused by non-uniformly distributed preferred directions, leaving cursor smoothing differences as the largest single algorithmic difference driving decoder performance.