Kalman filter control of a model of spatiotemporal cortical dynamics

Path integrals, particular kinds, and strange things

This paper describes a path integral formulation of the free energy principle. The ensuing account expresses the paths or trajectories that a particle takes as it evolves over time. The main results are a method or principle of least action that can be used to emulate the behaviour of particles in open exchange with their external milieu. Particles are defined by a particular partition, in which internal states are individuated from external states by active and sensory blanket states. The variational principle at hand allows one to interpret internal dynamics-of certain kinds of particles-as inferring external states that are hidden behind blanket states. We consider different kinds of particles, and to what extent they can be imbued with an elementary form of inference or sentience. Specifically, we consider the distinction between dissipative and conservative particles, inert and active particles and, finally, ordinary and strange particles. Strange particles can be described as inferring their own actions, endowing them with apparent autonomy or agency. In short-of the kinds of particles afforded by a particular partition-strange kinds may be apt for describing sentient behaviour.

An in-silico framework for modeling optimal control of neural systems

Introduction

Brain-machine interfaces have reached an unprecedented capacity to measure and drive activity in the brain, allowing restoration of impaired sensory, cognitive or motor function. Classical control theory is pushed to its limit when aiming to design control laws that are suitable for large-scale, complex neural systems. This work proposes a scalable, data-driven, unified approach to study brain-machine-environment interaction using established tools from dynamical systems, optimal control theory, and deep learning.

Methods

To unify the methodology, we define the environment, neural system, and prosthesis in terms of differential equations with learnable parameters, which effectively reduce to recurrent neural networks in the discrete-time case. Drawing on tools from optimal control, we describe three ways to train the system: Direct optimization of an objective function, oracle-based learning, and reinforcement learning. These approaches are adapted to different assumptions about knowledge of system equations, linearity, differentiability, and observability.

Results

We apply the proposed framework to train an in-silico neural system to perform tasks in a linear and a nonlinear environment, namely particle stabilization and pole balancing. After training, this model is perturbed to simulate impairment of sensor and motor function. We show how a prosthetic controller can be trained to restore the behavior of the neural system under increasing levels of perturbation.

Discussion

We expect that the proposed framework will enable rapid and flexible synthesis of control algorithms for neural prostheses that reduce the need for in-vivo testing. We further highlight implications for sparse placement of prosthetic sensor and actuator components.

Optimization of an unscented Kalman filter for an embedded platform

The unscented Kalman filter (UKF) is finding increased application in biological fields. While realizing a complex UKF system in a low-power embedded platform offers many potential benefits including wearability, it also poses significant design challenges. Here we present a method for optimizing a UKF system for realization in an embedded platform. The method seeks to minimize both computation time and error in UKF state reconstruction and forecasting. As a case study, we applied the method to a model for the rat sleep-wake regulatory system in which 432 variants of the UKF over six different variables are considered. The optimization method is divided into three stages that assess computation time, state forecast error, and state reconstruction error. We apply a cost function to variants that pass all three stages to identify a variant that computes 27 times faster than the reference variant and maintains required levels of state estimation and forecasting accuracy. We draw the following insights: 1) process noise provides leeway for simplifying the model and its integration in ways that speed computation time while maintaining state forecasting accuracy, 2) the assimilation of observed data during the UKF correction step provides leeway for simplifying the UKF structure in ways that speed computation time while maintaining state reconstruction accuracy, and 3) the optimization process can be accelerated by decoupling variables that directly impact the underlying model from variables that impact the UKF structure.

Optimizing deep brain stimulation based on isostable amplitude in essential tremor patient models

OBJECTIVE

Deep brain stimulation is a treatment for medically refractory essential tremor. To improve the therapy, closed-loop approaches are designed to deliver stimulation according to the system's state, which is constantly monitored by recording a pathological signal associated with symptoms (e.g. brain signal or limb tremor). Since the space of possible closed-loop stimulation strategies is vast and cannot be fully explored experimentally, how to stimulate according to the state should be informed by modeling. A typical modeling goal is to design a stimulation strategy that aims to maximally reduce the Hilbert amplitude of the pathological signal in order to minimize symptoms. Isostables provide a notion of amplitude related to convergence time to the attractor, which can be beneficial in model-based control problems. However, how isostable and Hilbert amplitudes compare when optimizing the amplitude response to stimulation in models constrained by data is unknown.

APPROACH

We formulate a simple closed-loop stimulation strategy based on models previously fitted to phase-locked deep brain stimulation data from essential tremor patients. We compare the performance of this strategy in suppressing oscillatory power when based on Hilbert amplitude and when based on isostable amplitude. We also compare performance to phase-locked stimulation and open-loop high-frequency stimulation.

MAIN RESULTS

For our closed-loop phase space stimulation strategy, stimulation based on isostable amplitude is significantly more effective than stimulation based on Hilbert amplitude when amplitude field computation time is limited to minutes. Performance is similar when there are no constraints, however constraints on computation time are expected in clinical applications. Even when computation time is limited to minutes, closed-loop phase space stimulation based on isostable amplitude is advantageous compared to phase-locked stimulation, and is more efficient than high-frequency stimulation.

SIGNIFICANCE

Our results suggest a potential benefit to using isostable amplitude more broadly for model-based optimization of stimulation in neurological disorders.

Wnt/β-catenin Signaling in Tissue Self-Organization

Across metazoans, animal body structures and tissues exist in robust patterns that arise seemingly out of stochasticity of a few early cells in the embryo. These patterns ensure proper tissue form and function during early embryogenesis, development, homeostasis, and regeneration. Fundamental questions are how these patterns are generated and maintained during tissue homeostasis and regeneration. Though fascinating scientists for generations, these ideas remain poorly understood. Today, it is apparent that the Wnt/β-catenin pathway plays a central role in tissue patterning. Wnt proteins are small diffusible morphogens which are essential for cell type specification and patterning of tissues. In this review, we highlight several mechanisms described where the spatial properties of Wnt/β-catenin signaling are controlled, allowing them to work in combination with other diffusible molecules to control tissue patterning. We discuss examples of this self-patterning behavior during development and adult tissues' maintenance. The combination of new physiological culture systems, mathematical approaches, and synthetic biology will continue to fuel discoveries about how tissues are patterned. These insights are critical for understanding the intricate interplay of core patterning signals and how they become disrupted in disease.

Controllability of Networks of Multiple Coupled Neural Populations: An Analytical Method for Neuromodulation's Feasibility

Neuromodulation plays a vital role in the prevention and treatment of neurological and psychiatric disorders. Neuromodulation's feasibility is a long-standing issue because it provides the necessity for neuromodulation to realize the desired purpose. A controllability analysis of neural dynamics is necessary to ensure neuromodulation's feasibility. Here, we present such a theoretical method by using the concept of controllability from the control theory that neuromodulation's feasibility can be studied smoothly. Firstly, networks of multiple coupled neural populations with different topologies are established to mathematically model complicated neural dynamics. Secondly, an analytical method composed of a linearization method, the Kalman controllable rank condition and a controllability index is applied to analyze the controllability of the established network models. Finally, the relationship between network dynamics or topological characteristic parameters and controllability is studied by using the analytical method. The proposed method provides a new idea for the study of neuromodulation's feasibility, and the results are expected to guide us to better modulate neurodynamics by optimizing network dynamics and network topology.

Seizure control in a neural mass model by an active disturbance rejection approach

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.

Calcium imaging and dynamic causal modelling reveal brain-wide changes in effective connectivity and synaptic dynamics during epileptic seizures

Pathophysiological explanations of epilepsy typically focus on either the micro/mesoscale (e.g. excitation-inhibition imbalance), or on the macroscale (e.g. network architecture). Linking abnormalities across spatial scales remains difficult, partly because of technical limitations in measuring neuronal signatures concurrently at the scales involved. Here we use light sheet imaging of the larval zebrafish brain during acute epileptic seizure induced with pentylenetetrazole. Spectral changes of spontaneous neuronal activity during the seizure are then modelled using neural mass models, allowing Bayesian inference on changes in effective network connectivity and their underlying synaptic dynamics. This dynamic causal modelling of seizures in the zebrafish brain reveals concurrent changes in synaptic coupling at macro- and mesoscale. Fluctuations of both synaptic connection strength and their temporal dynamics are required to explain observed seizure patterns. These findings highlight distinct changes in local (intrinsic) and long-range (extrinsic) synaptic transmission dynamics as a possible seizure pathomechanism and illustrate how our Bayesian model inversion approach can be used to link existing neural mass models of seizure activity and novel experimental methods.

Efficient implementation of a real-time estimation system for thalamocortical hidden Parkinsonian properties

Real-time estimation of dynamical characteristics of thalamocortical cells, such as dynamics of ion channels and membrane potentials, is useful and essential in the study of the thalamus in Parkinsonian state. However, measuring the dynamical properties of ion channels is extremely challenging experimentally and even impossible in clinical applications. This paper presents and evaluates a real-time estimation system for thalamocortical hidden properties. For the sake of efficiency, we use a field programmable gate array for strictly hardware-based computation and algorithm optimization. In the proposed system, the FPGA-based unscented Kalman filter is implemented into a conductance-based TC neuron model. Since the complexity of TC neuron model restrains its hardware implementation in parallel structure, a cost efficient model is proposed to reduce the resource cost while retaining the relevant ionic dynamics. Experimental results demonstrate the real-time capability to estimate thalamocortical hidden properties with high precision under both normal and Parkinsonian states. While it is applied to estimate the hidden properties of the thalamus and explore the mechanism of the Parkinsonian state, the proposed method can be useful in the dynamic clamp technique of the electrophysiological experiments, the neural control engineering and brain-machine interface studies.

Recovery of Dynamics and Function in Spiking Neural Networks with Closed-Loop Control

There is a growing interest in developing novel brain stimulation methods to control disease-related aberrant neural activity and to address basic neuroscience questions. Conventional methods for manipulating brain activity rely on open-loop approaches that usually lead to excessive stimulation and, crucially, do not restore the original computations performed by the network. Thus, they are often accompanied by undesired side-effects. Here, we introduce delayed feedback control (DFC), a conceptually simple but effective method, to control pathological oscillations in spiking neural networks (SNNs). Using mathematical analysis and numerical simulations we show that DFC can restore a wide range of aberrant network dynamics either by suppressing or enhancing synchronous irregular activity. Importantly, DFC, besides steering the system back to a healthy state, also recovers the computations performed by the underlying network. Finally, using our theory we identify the role of single neuron and synapse properties in determining the stability of the closed-loop system.

Brain stimulation is being used to ease symptoms in several neurological disorders in cases where pharmacological treatment is not effective (anymore). The most common way for stimulation so far has been to apply a fixed, predetermined stimulus irrespective of the actual state of the brain or the condition of the patient. Recently, alternative strategies such as event-triggered stimulation protocols have attracted the interest of researchers. In these protocols the state of the affected brain area is continuously monitored, but the stimulus is only applied if certain criteria are met. Here we go one step further and present a truly closed-loop stimulation protocol. That is, a stimulus is being continuously provided and the magnitude of the stimulus depends, at any point in time, on the ongoing neural activity dynamics of the affected brain area. This results not only in suppression of the pathological activity, but also in a partial recovery of the transfer function of the activity dynamics. Thus, the ability of the lesioned brain area to carry out relevant computations is restored up to a point as well.

Bidirectional interactions between neuronal and hemodynamic responses to transcranial direct current stimulation (tDCS): challenges for brain-state dependent tDCS

Transcranial direct current stimulation (tDCS) has been shown to modulate cortical neural activity. During neural activity, the electric currents from excitable membranes of brain tissue superimpose in the extracellular medium and generate a potential at scalp, which is referred as the electroencephalogram (EEG). Respective neural activity (energy demand) has been shown to be closely related, spatially and temporally, to cerebral blood flow (CBF) that supplies glucose (energy supply) via neurovascular coupling. The hemodynamic response can be captured by near-infrared spectroscopy (NIRS), which enables continuous monitoring of cerebral oxygenation and blood volume. This neurovascular coupling phenomenon led to the concept of neurovascular unit (NVU) that consists of the endothelium, glia, neurons, pericytes, and the basal lamina. Here, recent works suggest NVU as an integrated system working in concert using feedback mechanisms to enable proper brain homeostasis and function where the challenge remains in capturing these mostly nonlinear spatiotemporal interactions within NVU for brain-state dependent tDCS. In principal accordance, we propose EEG-NIRS-based whole-head monitoring of tDCS-induced neuronal and hemodynamic alterations during tDCS.

Dynamic causal modelling of electrographic seizure activity using Bayesian belief updating

Seizure activity in EEG recordings can persist for hours with seizure dynamics changing rapidly over time and space. To characterise the spatiotemporal evolution of seizure activity, large data sets often need to be analysed. Dynamic causal modelling (DCM) can be used to estimate the synaptic drivers of cortical dynamics during a seizure; however, the requisite (Bayesian) inversion procedure is computationally expensive. In this note, we describe a straightforward procedure, within the DCM framework, that provides efficient inversion of seizure activity measured with non-invasive and invasive physiological recordings; namely, EEG/ECoG. We describe the theoretical background behind a Bayesian belief updating scheme for DCM. The scheme is tested on simulated and empirical seizure activity (recorded both invasively and non-invasively) and compared with standard Bayesian inversion. We show that the Bayesian belief updating scheme provides similar estimates of time-varying synaptic parameters, compared to standard schemes, indicating no significant qualitative change in accuracy. The difference in variance explained was small (less than 5%). The updating method was substantially more efficient, taking approximately 5–10 min compared to approximately 1–2 h. Moreover, the setup of the model under the updating scheme allows for a clear specification of how neuronal variables fluctuate over separable timescales. This method now allows us to investigate the effect of fast (neuronal) activity on slow fluctuations in (synaptic) parameters, paving a way forward to understand how seizure activity is generated.

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We describe a DCM procedure that provides efficient inversion of seizure activity.

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Similar accuracy but substantially more efficient compared to standard DCM methods.

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Physiological fluctuations over different timescales can be specified.

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This scheme should contribute to understanding seizure activity using DCM.

Model-based estimation of intra-cortical connectivity using electrophysiological data

This paper provides a new method for model-based estimation of intra-cortical connectivity from electrophysiological measurements. A novel closed-form solution for the connectivity function of the Amari neural field equations is derived as a function of electrophysiological observations. The resultant intra-cortical connectivity estimate is driven from experimental data, but constrained by the mesoscopic neurodynamics that are encoded in the computational model. A demonstration is provided to show how the method can be used to image physiological mechanisms that govern cortical dynamics, which are normally hidden in clinical data from epilepsy patients. Accurate estimation performance is demonstrated using synthetic data. Following the computational testing, results from patient data are obtained that indicate a dominant increase in surround inhibition prior to seizure onset that subsides in the cases when the seizures spread.

Closed-loop and activity-guided optogenetic control

Advances in optical manipulation and observation of neural activity have set the stage for widespread implementation of closed-loop and activity-guided optical control of neural circuit dynamics. Closing the loop optogenetically (i.e., basing optogenetic stimulation on simultaneously observed dynamics in a principled way) is a powerful strategy for causal investigation of neural circuitry. In particular, observing and feeding back the effects of circuit interventions on physiologically relevant timescales is valuable for directly testing whether inferred models of dynamics, connectivity, and causation are accurate in vivo. Here we highlight technical and theoretical foundations as well as recent advances and opportunities in this area, and we review in detail the known caveats and limitations of optogenetic experimentation in the context of addressing these challenges with closed-loop optogenetic control in behaving animals.

Characterising seizures in anti-NMDA-receptor encephalitis with dynamic causal modelling

We characterised the pathophysiology of seizure onset in terms of slow fluctuations in synaptic efficacy using EEG in patients with anti-N-methyl-d-aspartate receptor (NMDA-R) encephalitis. EEG recordings were obtained from two female patients with anti-NMDA-R encephalitis with recurrent partial seizures (ages 19 and 31). Focal electrographic seizure activity was localised using an empirical Bayes beamformer. The spectral density of reconstructed source activity was then characterised with dynamic causal modelling (DCM). Eight models were compared for each patient, to evaluate the relative contribution of changes in intrinsic (excitatory and inhibitory) connectivity and endogenous afferent input. Bayesian model comparison established a role for changes in both excitatory and inhibitory connectivity during seizure activity (in addition to changes in the exogenous input). Seizures in both patients were associated with a sequence of changes in inhibitory and excitatory connectivity; a transient increase in inhibitory connectivity followed by a transient increase in excitatory connectivity and a final peak of excitatory–inhibitory balance at seizure offset. These systematic fluctuations in excitatory and inhibitory gain may be characteristic of (anti NMDA-R encephalitis) seizures. We present these results as a case study and replication to motivate analyses of larger patient cohorts, to see whether our findings generalise and further characterise the mechanisms of seizure activity in anti-NMDA-R encephalitis.

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We characterised seizures in patient with anti-NMDA-R encephalitis using EEG.

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Dynamic causal modelling was used to estimate causes of seizure activity.

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Characteristic variation of excitatory–inhibitory balance during seizure activity.

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This variation was seen for seizures within and between patients.

Estimation of effective connectivity via data-driven neural modeling

This research introduces a new method for functional brain imaging via a process of model inversion. By estimating parameters of a computational model, we are able to track effective connectivity and mean membrane potential dynamics that cannot be directly measured using electrophysiological measurements alone. The ability to track the hidden aspects of neurophysiology will have a profound impact on the way we understand and treat epilepsy. For example, under the assumption the model captures the key features of the cortical circuits of interest, the framework will provide insights into seizure initiation and termination on a patient-specific basis. It will enable investigation into the effect a particular drug has on specific neural populations and connectivity structures using minimally invasive measurements. The method is based on approximating brain networks using an interconnected neural population model. The neural population model is based on a neural mass model that describes the functional activity of the brain, capturing the mesoscopic biophysics and anatomical structure. The model is made subject-specific by estimating the strength of intra-cortical connections within a region and inter-cortical connections between regions using a novel Kalman filtering method. We demonstrate through simulation how the framework can be used to track the mechanisms involved in seizure initiation and termination.

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.

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.

Parameter estimation and control for a neural mass model based on the unscented Kalman filter

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.

Minimum energy control for in vitro neurons

OBJECTIVE

To demonstrate the applicability of optimal control theory for designing minimum energy charge-balanced input waveforms for single periodically-firing in vitro neurons from brain slices of Long-Evans rats.

APPROACH

The method of control uses the phase model of a neuron and does not require prior knowledge of the neuron's biological details. The phase model of a neuron is a one-dimensional model that is characterized by the neuron's phase response curve (PRC), a sensitivity measure of the neuron to a stimulus applied at different points in its firing cycle. The PRC for each neuron is experimentally obtained by measuring the shift in phase due to a short-duration pulse injected into the periodically-firing neuron at various phase values. Based on the measured PRC, continuous-time, charge-balanced, minimum energy control waveforms have been designed to regulate the next firing time of the neuron upon application at the onset of an action potential.

MAIN RESULT

The designed waveforms can achieve the inter-spike-interval regulation for in vitro neurons with energy levels that are lower than those of conventional monophasic pulsatile inputs of past studies by at least an order of magnitude. They also provide the advantage of being charge-balanced. The energy efficiency of these waveforms is also shown by performing several supporting simulations that compare the performance of the designed waveforms against that of phase shuffled surrogate inputs, variants of the minimum energy waveforms obtained from suboptimal PRCs, as well as pulsatile stimuli that are applied at the point of maximum PRC. It was found that the minimum energy waveforms perform better than all other stimuli both in terms of control and in the amount of energy used. Specifically, it was seen that these charge-balanced waveforms use at least an order of magnitude less energy than conventional monophasic pulsatile stimuli.

SIGNIFICANCE

The significance of this work is that it uses concepts from the theory of optimal control and introduces a novel approach in designing minimum energy charge-balanced input waveforms for neurons that are robust to noise and implementable in electrophysiological experiments.

The role of inhibition in oscillatory wave dynamics in the cortex

Cortical oscillations arise during behavioral and mental tasks, and all temporal oscillations have particular spatial patterns. Studying the mechanisms that generate and modulate the spatiotemporal characteristics of oscillations is important for understanding neural information processing and the signs and symptoms of dynamical diseases of the brain. Nevertheless, it remains unclear how GABAergic inhibition modulates these oscillation dynamics. Using voltage-sensitive dye imaging, pharmacological methods, and tangentially cut occipital neocortical brain slices (including layers 3-5) of Sprague-Dawley rat, we found that GABAa disinhibition with bicuculline can progressively simplify oscillation dynamics in the presence of carbachol in a concentration-dependent manner. Additionally, GABAb disinhibition can further simplify oscillation dynamics after GABAa receptors are blocked. Both GABAa and GABAb disinhibition increase the synchronization of the neural network. Theta frequency (5-15-Hz) oscillations are reliably generated by using a combination of GABAa and GABAb antagonists alone. These theta oscillations have basic spatiotemporal patterns similar to those generated by carbachol/bicuculline. These results are illustrative of how GABAergic inhibition increases the complexity of patterns of activity and contributes to the regulation of the cortex.

Dynamic causal modelling of lateral interactions in the visual cortex

This paper presents a dynamic causal model based upon neural field models of the Amari type. We consider the application of these models to non-invasive data, with a special focus on the mapping from source activity on the cortical surface to a single channel. We introduce a neural field model based upon the canonical microcircuit (CMC), in which neuronal populations are assigned to different cortical layers. We show that DCM can disambiguate between alternative (neural mass and field) models of cortical activity. However, unlike neural mass models, DCM with neural fields can address questions about neuronal microcircuitry and lateral interactions. This is because they are equipped with interlaminar connections and horizontal intra-laminar connections that are patchy in nature. These horizontal or lateral connections can be regarded as connecting macrocolumns with similar feature selectivity. Crucially, the spatial parameters governing horizontal connectivity determine the separation (width) of cortical macrocolumns. Thus we can estimate the width of macro columns, using non-invasive electromagnetic signals. We illustrate this estimation using dynamic causal models of steady-state or ongoing spectral activity measured using magnetoencephalography (MEG) in human visual cortex. Specifically, we revisit the hypothesis that the size of a macrocolumn is a key determinant of neuronal dynamics, particularly the peak gamma frequency. We are able to show a correlation, over subjects, between columnar size and peak gamma frequency — that fits comfortably with established correlations between peak gamma frequency and the size of visual cortex defined retinotopically. We also considered cortical excitability and assessed its relative influence on observed gamma activity. This example highlights the potential utility of dynamic causal modelling and neural fields in providing quantitative characterisations of spatially extended dynamics on the cortical surface — that are parameterised in terms of horizontal connections, implicit in the cortical micro-architecture and its synaptic parameters.

Spatiotemporal multi-resolution approximation of the Amari type neural field model

Neural fields are spatially continuous state variables described by integro-differential equations, which are well suited to describe the spatiotemporal evolution of cortical activations on multiple scales. Here we develop a multi-resolution approximation (MRA) framework for the integro-difference equation (IDE) neural field model based on semi-orthogonal cardinal B-spline wavelets. In this way, a flexible framework is created, whereby both macroscopic and microscopic behavior of the system can be represented simultaneously. State and parameter estimation is performed using the expectation maximization (EM) algorithm. A synthetic example is provided to demonstrate the framework.

Using "smart stimulators" to treat Parkinson's disease: re-engineering neurostimulation devices

Let's imagine the cruise control of your car locked at 120 km/h on any road in any condition (city, country, highway, sunny or rainy weather), or your car air conditioner set on maximum cold in any temperature condition (even during a snowy winter): would you find it efficient? That would probably not be the most optimal strategy for a proper and comfortable driving experience. As surprising as this may seem, this is a pretty accurate illustration of how deep brain stimulation is used today to treat Parkinson's disease motor symptoms and other neurological disorders such as essential tremor, obsessive-compulsive disorder, or epilepsy.

Minimum energy desynchronizing control for coupled neurons

We employ optimal control theory to design an event-based, minimum energy, desynchronizing control stimulus for a network of pathologically synchronized, heterogeneously coupled neurons. This works by optimally driving the neurons to their phaseless sets, switching the control off, and letting the phases of the neurons randomize under intrinsic background noise. An event-based minimum energy input may be clinically desirable for deep brain stimulation treatment of neurological diseases, like Parkinson's disease. The event-based nature of the input results in its administration only when it is necessary, which, in general, amounts to fewer applications, and hence, less charge transfer to and from the tissue. The minimum energy nature of the input may also help prolong battery life for implanted stimulus generators. For the example considered, it is shown that the proposed control causes a considerable amount of randomization in the timing of each neuron's next spike, leading to desynchronization for the network.

Estimating the unmeasured membrane potential of neuronal populations from the EEG using a class of deterministic nonlinear filters

We present a model-based estimation method to reconstruct the unmeasured membrane potential of neuronal populations from a single-channel electroencephalographic (EEG) measurement. We consider a class of neural mass models that share a general structure, specifically the models by Stam et al (1999 Clin. Neurophysiol. 110 1801-13), Jansen and Rit (1995 Biol. Cybern. 73 357-66) and Wendling et al (2005 J. Clin. Neurophysiol. 22 343). Under idealized assumptions, we prove the global exponential convergence of our filter. Then, under more realistic assumptions, we investigate the robustness of our filter against model uncertainties and disturbances. Analytic proofs are provided for all results and our analyses are further illustrated via simulations.

Canards of mixed type in a neural burster

Canards are solutions of slow-fast systems that spend long times near branches of repelling equilibria, periodic orbits, or higher-dimensional invariant sets. Here, we report on the observation of a new type of canard orbit, labeled a canard of mixed type. This canard orbit is a hybrid of the classical limit cycle canards, which spend long times near attracting and repelling branches of equilibria, and torus canards, which spend long times near attracting and repelling branches of periodic orbits. The canards of mixed type arise in a model of neural bursting activity of fold-fold cycle type, and, as other canard phenomena, separate different dynamic states.

Evidence-based modeling of network discharge dynamics during periodic pacing to control epileptiform activity

Deep brain stimulation (DBS) is a promising therapeutic approach for epilepsy treatment. Recently, research has focused on the implementation of stimulation protocols that would adapt to the patients need (adaptive stimulation) and deliver electrical stimuli only when it is most useful. A formal mathematical description of the effects of electrical stimulation on neuronal networks is a prerequisite for the development of adaptive DBS algorithms. Using tools from non-linear dynamic analysis, we describe an evidence-based, mathematical modeling approach that (1) accurately simulates epileptiform activity at time-scales of single and multiple ictal discharges, (2) simulates modulation of neural dynamics during epileptiform activity in response to fixed, low-frequency electrical stimulation, (3) defines a mapping from real-world observations to model state, and (4) defines a mapping from model state to real-world observations. We validate the real-world utility of the model's properties by statistical comparison between the number, duration, and interval of ictal-like discharges observed in vitro and those simulated in silica under conditions of repeated stimuli at fixed-frequency. These validation results confirm that the evidence-based modeling approach captures robust, informative features of neural network dynamics of in vitro epileptiform activity under periodic pacing and support its use for further implementation of adaptive DBS protocols for epilepsy treatment.

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.

Dynamic causal modeling with neural fields

The aim of this paper is twofold: first, to introduce a neural field model motivated by a well-known neural mass model; second, to show how one can estimate model parameters pertaining to spatial (anatomical) properties of neuronal sources based on EEG or LFP spectra using Bayesian inference. Specifically, we consider neural field models of cortical activity as generative models in the context of dynamic causal modeling (DCM). This paper considers the simplest case of a single cortical source modeled by the spatiotemporal dynamics of hidden neuronal states on a bounded cortical surface or manifold. We build this model using multiple layers, corresponding to cortical lamina in the real cortical manifold. These layers correspond to the populations considered in classical (Jansen and Rit) neural mass models. This allows us to formulate a neural field model that can be reduced to a neural mass model using appropriate constraints on its spatial parameters. In turn, this enables one to compare and contrast the predicted responses from equivalent neural field and mass models respectively. We pursue this using empirical LFP data from a single electrode to show that the parameters controlling the spatial dynamics of cortical activity can be recovered, using DCM, even in the absence of explicit spatial information in observed data.

Exploring optimal current stimuli that provide membrane voltage tracking in a neuron model

Studying neurons from an energy efficiency perspective has produced results in the research literature. This paper presents a method that enables computation of low energy input current stimuli that are able to drive a reduced Hodgkin-Huxley neuron model to approximate a prescribed time-varying reference membrane voltage. An optimal control technique is used to discover an input current that optimally minimizes a user selected balance between the square of the input stimulus current (input current 'energy') and the difference between the reference voltage and the membrane voltage (tracking error) over a stimulation period. Selecting reference signals to be membrane voltages produced by the neuron model in response to common types of input currents i(t) enables a comparison between i(t) and the determined optimal current stimulus i*(t). The intent is not to modify neuron dynamics, but through comparison of i(t) and i*(t) provide insight into neuron dynamics. Simulation results for four different bifurcation types demonstrate that this method consistently finds lower energy stimulus currents i*(t) that are able to approximate membrane voltages as produced by higher energy input currents i(t) in this neuron model.

A data-driven framework for neural field modeling

This paper presents a framework for creating neural field models from electrophysiological data. The Wilson and Cowan or Amari style neural field equations are used to form a parametric model, where the parameters are estimated from data. To illustrate the estimation framework, data is generated using the neural field equations incorporating modeled sensors enabling a comparison between the estimated and true parameters. To facilitate state and parameter estimation, we introduce a method to reduce the continuum neural field model using a basis function decomposition to form a finite-dimensional state-space model. Spatial frequency analysis methods are introduced that systematically specify the basis function configuration required to capture the dominant characteristics of the neural field. The estimation procedure consists of a two-stage iterative algorithm incorporating the unscented Rauch-Tung-Striebel smoother for state estimation and a least squares algorithm for parameter estimation. The results show that it is theoretically possible to reconstruct the neural field and estimate intracortical connectivity structure and synaptic dynamics with the proposed framework.

Model-driven therapeutic treatment of neurological disorders: reshaping brain rhythms with neuromodulation

Electric stimulation has been investigated for several decades to treat, with various degrees of success, a broad spectrum of neurological disorders. Historically, the development of these methods has been largely empirical but has led to a remarkably efficient, yet invasive treatment: deep brain stimulation (DBS). However, the efficiency of DBS is limited by our lack of understanding of the underlying physiological mechanisms and by the complex relationship existing between brain processing and behaviour. Biophysical modelling of brain activity, describing multi-scale spatio-temporal patterns of neuronal activity using a mathematical model and taking into account the physical properties of brain tissue, represents one way to fill this gap. In this review, we illustrate how biophysical modelling is beginning to emerge as a driving force orienting the development of innovative brain stimulation methods that may move DBS forward. We present examples of modelling works that have provided fruitful insights in regards to DBS underlying mechanisms, and others that also suggest potential improvements for this neurosurgical procedure. The reviewed literature emphasizes that biophysical modelling is a valuable tool to assist a rational development of electrical and/or magnetic brain stimulation methods tailored to both the disease and the patient's characteristics.

Towards model-based control of Parkinson's disease

Modern model-based control theory has led to transformative improvements in our ability to track the nonlinear dynamics of systems that we observe, and to engineer control systems of unprecedented efficacy. In parallel with these developments, our ability to build computational models to embody our expanding knowledge of the biophysics of neurons and their networks is maturing at a rapid rate. In the treatment of human dynamical disease, our employment of deep brain stimulators for the treatment of Parkinson's disease is gaining increasing acceptance. Thus, the confluence of these three developments--control theory, computational neuroscience and deep brain stimulation--offers a unique opportunity to create novel approaches to the treatment of this disease. This paper explores the relevant state of the art of science, medicine and engineering, and proposes a strategy for model-based control of Parkinson's disease. We present a set of preliminary calculations employing basal ganglia computational models, structured within an unscented Kalman filter for tracking observations and prescribing control. Based upon these findings, we will offer suggestions for future research and development.

Assimilating seizure dynamics

Observability of a dynamical system requires an understanding of its state-the collective values of its variables. However, existing techniques are too limited to measure all but a small fraction of the physical variables and parameters of neuronal networks. We constructed models of the biophysical properties of neuronal membrane, synaptic, and microenvironment dynamics, and incorporated them into a model-based predictor-controller framework from modern control theory. We demonstrate that it is now possible to meaningfully estimate the dynamics of small neuronal networks using as few as a single measured variable. Specifically, we assimilate noisy membrane potential measurements from individual hippocampal neurons to reconstruct the dynamics of networks of these cells, their extracellular microenvironment, and the activities of different neuronal types during seizures. We use reconstruction to account for unmeasured parts of the neuronal system, relating micro-domain metabolic processes to cellular excitability, and validate the reconstruction of cellular dynamical interactions against actual measurements. Data assimilation, the fusing of measurement with computational models, has significant potential to improve the way we observe and understand brain dynamics.

Kalman meets neuron: the emerging intersection of control theory with neuroscience

Since the 1950s, we have developed mature theories of modern control theory and computational neuroscience with almost no interaction between these disciplines. With the advent of computationally efficient nonlinear Kalman filtering techniques, along with improved neuroscience models that provide increasingly accurate reconstruction of dynamics in a variety of important normal and disease states in the brain, the prospects for a synergistic interaction between these fields are now strong. I show recent examples of the use of nonlinear control theory for the assimilation and control of single neuron dynamics, the modulation of oscillatory wave dynamics in brain cortex, a control framework for Parkinsonian dynamics and seizures, and the use of optimized parameter model networks to assimilate complex network data - the 'consensus set'.

A model of feedback control for the charge-balanced suppression of epileptic seizures

Here we present several refinements to a model of feedback control for the suppression of epileptic seizures. We utilize a stochastic partial differential equation (SPDE) model of the human cortex. First, we verify the strong convergence of numerical solutions to this model, paying special attention to the sharp spatial changes that occur at electrode edges. This allows us to choose appropriate step sizes for our simulations; because the spatial step size must be small relative to the size of an electrode in order to resolve its electrical behavior, we are able to include a more detailed electrode profile in the simulation. Then, based on evidence that the mean soma potential is not the variable most closely related to the measurement of a cortical surface electrode, we develop a new model for this. The model is based on the currents flowing in the cortex and is used for a simulation of feedback control. The simulation utilizes a new control algorithm incorporating the total integral of the applied electrical potential. Not only does this succeed in suppressing the seizure-like oscillations, but it guarantees that the applied signal will be charge-balanced and therefore unlikely to cause cortical damage.

Toward rational design of electrical stimulation strategies for epilepsy control

Electrical stimulation is emerging as a viable alternative for patients with epilepsy whose seizures are not alleviated by drugs or surgery. Its attractions are temporal and spatial specificity of action, flexibility of waveform parameters and timing, and the perception that its effects are reversible unlike resective surgery. However, despite significant advances in our understanding of mechanisms of neural electrical stimulation, clinical electrotherapy for seizures relies heavily on empirical tuning of parameters and protocols. We highlight concurrent treatment goals with potentially conflicting design constraints that must be resolved when formulating rational strategies for epilepsy electrotherapy, namely, seizure reduction versus cognitive impairment, stimulation efficacy versus tissue safety, and mechanistic insight versus clinical pragmatism. First, treatment markers, objectives, and metrics relevant to electrical stimulation for epilepsy are discussed from a clinical perspective. Then the experimental perspective is presented, with the biophysical mechanisms and modalities of open-loop electrical stimulation, and the potential benefits of closed-loop control for epilepsy.

Zwanzig-Mori projection operators and EEG dynamics: deriving a simple equation of motion

We present a macroscopic theory of electroencephalogram (EEG) dynamics based on the laws of motion that govern atomic and molecular motion. The theory is an application of Zwanzig-Mori projection operators. The result is a simple equation of motion that has the form of a generalized Langevin equation (GLE), which requires knowledge only of macroscopic properties. The macroscopic properties can be extracted from experimental data by one of two possible variational principles. These variational principles are our principal contribution to the formalism. Potential applications are discussed, including applications to the theory of critical phenomena in the brain, Granger causality and Kalman filters.

PACS code: 87.19.lj

Data assimilation for heterogeneous networks: the consensus set

Data assimilation in dynamical networks is intrinsically challenging. A method is introduced for the tracking of heterogeneous networks of oscillators or excitable cells in a nonstationary environment, using a homogeneous model network to expedite the accurate reconstruction of parameters and unobserved variables. An implementation using ensemble Kalman filtering to track the states of the heterogeneous network is demonstrated on simulated data and applied to a mammalian brain network experiment. The approach has broad applicability for the prediction and control of biological and physical networks.

Tracking and control of neuronal Hodgkin-Huxley dynamics

Nonlinear ensemble state estimation offers a paradigm-shifting improvement in our ability to observe, predict, and control the state of spiking neuronal systems. We use an ensemble Kalman filter to predict hidden states and future trajectories in the Hodgkin-Huxley equations, reconstruct ion dynamics, control neuronal activity including a strategy for dynamic conductance clamping, and show the feasibility of controlling pathological cellular activity such as seizures.

Hierarchical models in the brain

This paper describes a general model that subsumes many parametric models for continuous data. The model comprises hidden layers of state-space or dynamic causal models, arranged so that the output of one provides input to another. The ensuing hierarchy furnishes a model for many types of data, of arbitrary complexity. Special cases range from the general linear model for static data to generalised convolution models, with system noise, for nonlinear time-series analysis. Crucially, all of these models can be inverted using exactly the same scheme, namely, dynamic expectation maximization. This means that a single model and optimisation scheme can be used to invert a wide range of models. We present the model and a brief review of its inversion to disclose the relationships among, apparently, diverse generative models of empirical data. We then show that this inversion can be formulated as a simple neural network and may provide a useful metaphor for inference and learning in the brain.

Models are essential to make sense of scientific data, but they may also play a central role in how we assimilate sensory information. In this paper, we introduce a general model that generates or predicts diverse sorts of data. As such, it subsumes many common models used in data analysis and statistical testing. We show that this model can be fitted to data using a single and generic procedure, which means we can place a large array of data analysis procedures within the same unifying framework. Critically, we then show that the brain has, in principle, the machinery to implement this scheme. This suggests that the brain has the capacity to analyse sensory input using the most sophisticated algorithms currently employed by scientists and possibly models that are even more elaborate. The implications of this work are that we can understand the structure and function of the brain as an inference machine. Furthermore, we can ascribe various aspects of brain anatomy and physiology to specific computational quantities, which may help understand both normal brain function and how aberrant inferences result from pathological processes associated with psychiatric disorders.