Statistical Performance Analysis of Data-Driven Neural Models

Several studies posit energy as a constraint on the coding and processing of information in the brain due to the high cost of resting and evoked cortical activity. This suggestion has been addressed theoretically with models of a single neuron and two coupled neurons. Neural mass models (NMMs) address mean-field based modeling of the activity and interactions between populations of neurons rather than a few neurons. NMMs have been widely employed for studying the generation of EEG rhythms, and more recently as frameworks for integrated models of neurophysiology and functional MRI (fMRI) responses. To date, the consequences of energy constraints on the activity and interactions of ensembles of neurons have not been addressed. Here we aim to study the impact of constraining energy consumption during the resting-state on NMM parameters. To this end, we first linearized the model, then used stochastic control theory by introducing a quadratic cost function, which transforms the NMM into a stochastic linear quadratic regulator (LQR). Solving the LQR problem introduces a regime in which the NMM parameters, specifically the effective connectivities between neuronal populations, must vary with time. This is in contrast to current NMMs, which assume a constant parameter set for a given condition or task. We further simulated energy-constrained stochastic control of a specific NMM, the Wilson and Cowan model of two coupled neuronal populations, one of which is excitatory and the other inhibitory. These simulations demonstrate that with varying weights of the energy-cost function, the NMM parameters show different time-varying behavior. We conclude that constraining NMMs according to energy consumption may create more realistic models. We further propose to employ linear NMMs with time-varying parameters as an alternative to traditional nonlinear NMMs with constant parameters.

A computational algorithm is presented for the Bayesian Cramer-Rao lower bound (BCRB) in filtering applications with measurement noise from mixture distributions with jump Markov switching structure. Such mixture distributions are common for radio propagation in mixed line- and non-line-of-sight environments. The newly derived BCRB is tighter than earlier more general bounds proposed in literature, and thus gives a more realistic bound on actual estimation performance. The resulting BCRB can be used to compute a lower bound on root mean square error of position estimates in a large class of radio localization applications. We illustrate this on an archetypical tracking application using a nearly constant velocity model and time of arrival observations.

Neural mass models are used to simulate cortical dynamics and to explain the electrical and magnetic fields measured using electro- and magnetoencephalography. Simulations evince a complex phase-space structure for these kinds of models; including stationary points and limit cycles and the possibility for bifurcations and transitions among different modes of activity. This complexity allows neural mass models to describe the itinerant features of brain dynamics. However, expressive, nonlinear neural mass models are often difficult to fit to empirical data without additional simplifying assumptions: e.g., that the system can be modelled as linear perturbations around a fixed point. In this study we offer a mathematical analysis of neural mass models, specifically the canonical microcircuit model, providing analytical solutions describing slow changes in the type of cortical activity, i.e. dynamical itinerancy. We derive a perturbation analysis up to second order of the phase flow, together with adiabatic approximations. This allows us to describe amplitude modulations in a relatively simple mathematical format providing analytic proof-of-principle for the existence of semi-stable states of cortical dynamics at the scale of a cortical column. This work allows for model inversion of neural mass models, not only around fixed points, but over regions of phase space that encompass transitions among semi or multi-stable states of oscillatory activity. Crucially, these theoretical results speak to model inversion in the context of multiple semi-stable brain states, such as the transition between interictal, pre-ictal and ictal activity in epilepsy.

Combining the neural mass model and Hodgkin–Huxley formalism: Neuronal dynamics modelling

Recent progress in Kalman filters to estimate states and parameters in nonlinear systems has provided the possibility of applying such approaches to neural systems. We here apply the nonlinear method of unscented Kalman filters (UKFs) to observe states and estimate parameters in a neural mass model that can simulate distinct rhythms in electroencephalography (EEG) including dynamical evolution during epilepsy seizures. We demonstrate the efficiency of the UKF in estimating states and parameters. We also develop an UKF-based control strategy to modulate the dynamics of the neural mass model. In this strategy the UKF plays the role of observing states, and the control law is constructed via the estimated states. We demonstrate the feasibility of using such a strategy to suppress epileptiform spikes in the neural mass model.

This paper addresses the problem of estimating time varying phase noise caused by imperfect oscillators in multiple-input multiple-output (MIMO) systems. The estimation problem is parameterized in detail and based on an equivalent signal model its dimensionality is reduced to minimize the overhead associated with phase noise estimation. New exact and closed-form expressions for the Bayesian Cramer-Rao lower bounds (BCRLBs) and soft-input maximum a posteriori (MAP) estimators for online, i.e., filtering, and offline, i.e., smoothing, estimation of phase noise over the length of a frame are derived. Simulations demonstrate that the proposed MAP estimators' mean-square error (MSE) performances are very close to the derived BCRLBs at moderate-to-high signal-to-noise ratios. To reduce the overhead and complexity associated with tracking the phase noise processes over the length of a frame, a novel soft-input extended Kalman filter (EKF) and extended Kalman smoother (EKS) that use soft statistics of the transmitted symbols given the current observations are proposed. Numerical results indicate that by employing the proposed phase tracking approach, the bit-error rate performance of a MIMO system affected by phase noise can be significantly improved. In addition, simulation results indicate that the proposed phase noise estimation scheme allows for application of higher order modulations and larger numbers of antennas in MIMO systems that employ imperfect oscillators.

In this paper, a joint revisit and dwell time management (JRDTM) strategy for single target tracking based on the predicted Bayesian Cramer-Rao lower bound (BCRLB) in phased array radar system is addressed. We achieve the time resources management by formulating and solving an optimization problem, which is to minimize the resource amount used for tracking with the tracking accuracy of the target meeting a predesigned threshold. As the BCRLB provides a lower bound on the estimated mean square error (MSE) of target state, the predicted BCRLB model with time resources is derived and employed as the metric for the tracking performance. We put forward a converted algorithm to settle the optimization problem subsequently. Simulation results demonstrate that the proposed joint management strategy can help to achieve the tracking performance with less resource consumed.

Electroencephalogram (EEG) and local field potential (LFP) signals are measured for both experimental and clinical purposes which include sleep stage analyses, brain–computer interfaces, and disease diagnosis. EEG and LFP data analyses are typically based on models assuming that the measured data is generated from a biological system and estimate the model parameter values that most accurately reproduce the measured data. Thus, use of a biologically plausible model is important for a model‐based analysis. However, analyses using models that include time delay and nonlinearity have not been reported, even though they are biologically important for EEG generation mechanisms. In this study, we developed a parameter estimation method that uses a particle filter for models with time delay and nonlinearity, which was evaluated with simulations. Simulated EEG data were generated from neural mass models (NMMs). The NMM parameters were estimated from the generated data. Furthermore, parameters for modeling EEG features of patients with Alzheimer's disease were included in the NMM; the disease parameters were estimated from the simulated EEG data. We observed that NMM parameters, as well as the disease parameters, were accurately estimated from the simulated data. We conclude that the validity of our method for estimating parameters of NMMs including time delay and nonlinearity is confirmed for simulated EEG data, and these results show the possibility of using our method for model‐based analysis with real EEG data. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

In this paper, the problem of controlling epileptiform spikes in a neural mass model is addressed. Considering the complication and nonlinearity of the neural mass model, a fuzzy PID controller is designed so that epileptiform spikes are quenched and the output waveform tracks an expected one. The tracking effect is analyzed by numerical simulation for a regular network of coupled neural populations. The effect of important model parameters on the control energy and the effect of the types of controlled populations on the ability to realize the tracking purpose are analyzed for the same network.

Neural mass models provide an attractive framework for modeling complex behavior in cortical circuits. The models are based on describing the dynamics of large neural populations through the space and time evolution of a small number of key aggregate statistical quantities. Fitting these models to electrode array recordings can provide insight into connectivity and structure of neural circuits as well as the response of these circuits to stimuli. However, neural mass models are fundamentally nonlinear dynamical systems with large numbers of hidden states, and validating the models on actual recordings and estimating the key parameters remains challenging. This work proposes a novel method for systematically identifying neural mass models that is particularly well-suited for high-density micro-electrocorticographic (μECoG) data. The methodology requires minimal assumptions on the model, and can automatically uncover the underlying components in the neural populations We discuss possible applications to in vivo recordings from feline visual cortex using a recently-developed, high-density 360 contact flexible electrode array with 500 µm inter-electrode spacing.

This paper introduces methods and a novel toolbox that efficiently integrates high-dimensional Neural Mass Models (NMMs) specified by two essential components. The first is the set of nonlinear Random Differential Equations (RDEs) of the dynamics of each neural mass. The second is the highly sparse three-dimensional Connectome Tensor (CT) that encodes the strength of the connections and the delays of information transfer along the axons of each connection. To date, simplistic assumptions prevail about delays in the CT, often assumed to be Dirac-delta functions. In reality, delays are distributed due to heterogeneous conduction velocities of the axons connecting neural masses. These distributed-delay CTs are challenging to model. Our approach implements these models by leveraging several innovations. Semi-analytical integration of RDEs is done with the Local Linearization (LL) scheme for each neural mass, ensuring dynamical fidelity to the original continuous-time nonlinear dynamic. This semi-analytic LL integration is highly computationally-efficient. In addition, a tensor representation of the CT facilitates parallel computation. It also seamlessly allows modeling distributed delays CT with any level of complexity or realism. This ease of implementation includes models with distributed-delay CTs. Consequently, our algorithm scales linearly with the number of neural masses and the number of equations they are represented with, contrasting with more traditional methods that scale quadratically at best. To illustrate the toolbox's usefulness, we simulate a single Zetterberg-Jansen and Rit (ZJR) cortical column, a single thalmo-cortical unit, and a toy example comprising 1000 interconnected ZJR columns. These simulations demonstrate the consequences of modifying the CT, especially by introducing distributed delays. The examples illustrate the complexity of explaining EEG oscillations, e.g., split alpha peaks, since they only appear for distinct neural masses. We provide an open-source Script for the toolbox.

Neural mass models (NMM) describe neural activity on a macroscopic scale, which can be compared to the electroencephalogram (EEG). This allows a better understanding of the processes responsible for various EEG patterns, including pathological rhythms as diffuse slowing or burst-suppression [1]. Using available models which contain explicit expressions for the synaptic response and number of synapses [2], pathological conditions that modulate synaptic function, such as anesthetics [3] and hypoxia, can be included easily. However, it is less obvious how to incorporate conditions which alter the excitability of neurons, such as hyperkalemia or channel blockers. Here, we present a method for constructing a neural mass model by using the relation between synaptic input of a single cell model and its firing rate. This allows an easy implementation for pathological conditions. We describe the average firing rate of a single population of neurons receiving one type of synaptic input, but this can readily be extended to multiple populations. A set of differential equations describes, traditionally, the average synaptic conductance [2]. Assuming Poisson statistics for the input, we can derive another equation, which describes the time evolution of the standard deviation of the synaptic conductance across the population. The average and standard deviation of the conductance then determine the distribution and the corresponding average of the firing rates in the population. As initial verification, the constructed mean field model is numerically compared to a network of single cells. From the single cell model we determine the dependence of the firing rate on (constant) synaptic conductance numerically. Furthermore, we show that, for fluctuating inputs, the firing rate is well approximated by the instantaneous synaptic conductance. 120 Hodgkin-Huxley type cells were connected all-to-all with inhibitory synapses: a simple configuration which results in intrinsic oscillations. Each cell receives inhibitory external input as well, consisting of Poisson trains. We find a close agreement between the constructed neural mass model and the network simulation (Figure 1). Figure 1. Comparison of step response of the derived NMM and a detailed network model The proposed method can easily be extended to model heterogeneous populations, multiple types of synapses, spatial structures, propagation delays, and bursting dynamics [4]. Any pathophysiology can readily be incorporated by adapting the single cell model. This allows for testing hypotheses on processes underlying abnormal EEGs.

In this Paper, to develop a new detection and control strategy for epilepsy, a multi -coupled neural mass model (NMM) is proposed to reproduce epileptic seizures. The unscented Kalman filter (UKF) based on bayesian estimation is used to estimate the states and parameters of the NMM and reconstruct normal and epilepsy brain states. Then, the closed-loop control system includes a computation model to reconstruct, a state observer to observe and a controller to adjust the brain activities depends on the slow permittivity variable in NMM.

In an effort to understand basic functional mechanisms that can produce epileptic seizures, we introduce some key features in a model of coupled neural populations that enable the generation of seizure-like events and similar dynamics with the ones observed during the route of the epileptic brain towards real seizures. In this model, modified from David and Friston’s neural mass model, an internal feedback mechanism is incorporated to maintain synchronous behavior within normal levels despite elevated coupling. Normal internal feedback quickly regulates an abnormally high coupling between the neural populations, whereas pathological internal feedback can lead to hypersynchronization and the appearance of seizure-like high amplitude oscillations. Feedback decoupling is introduced as a robust seizure control strategy. An external feedback decoupling controller is introduced to maintain normal synchronous behavior. The results from the analysis in this model have an interesting physical interpretation and specific implications for the treatment of epileptic seizures. The proposed model and control scheme are consistent with a variety of recent observations in the human and animal epileptic brain, and with theories from nonlinear systems, adaptive systems, optimization, and neurophysiology.

A closed-loop neuromodulation automatically adjusts stimuli according to brain response in real time. It is viewed as a promising way to control medically intractable epilepsy. A suitable closed-loop modulation strategy, which is robust enough to unknown nonlinearities, dynamics, and disturbances, is in great need in the clinic. For the specialization of epilepsy, the Jansen’s neural mass model is utilized to simulate the undesired high amplitudes epileptic activities, and active disturbance rejection control is designed to suppress the high amplitudes of epileptiform discharges. With the help of active disturbance rejection control, closed-loop roots of the system are far from the imaginary axis. Time domain response shows that active disturbance rejection control is able to control seizure no matter whether disturbances exist or not. At the same time, frequency domain response presents that enough stability margins and a broader range of tunable controller parameters can be obtained. Stable regions have also been presented to provide guidance to choose the parameters of active disturbance rejection control. Numerical results show that, compared with proportional-integral control, more accurate modulation with less energy can be achieved by active disturbance rejection control. It confirms that the active disturbance rejection control-based neuromodulation solution is able to achieve a desired performance. It is a promising closed-loop neuromodulation strategy in seizure control.