Minimum energy desynchronizing control for coupled neurons

Disrupting abnormal neuronal oscillations with adaptive delayed feedback control

Closed-loop neuronal stimulation has a strong therapeutic potential for neurological disorders such as Parkinson's disease. However, at the moment, standard stimulation protocols rely on continuous open-loop stimulation and the design of adaptive controllers is an active field of research. Delayed feedback control (DFC), a popular method used to control chaotic systems, has been proposed as a closed-loop technique for desynchronisation of neuronal populations but, so far, was only tested in computational studies. We implement DFC for the first time in neuronal populations and access its efficacy in disrupting unwanted neuronal oscillations. To analyse in detail the performance of this activity control algorithm, we used specialised in vitro platforms with high spatiotemporal monitoring/stimulating capabilities. We show that the conventional DFC in fact worsens the neuronal population oscillatory behaviour, which was never reported before. Conversely, we present an improved control algorithm, adaptive DFC (aDFC), which monitors the ongoing oscillation periodicity and self-tunes accordingly. aDFC effectively disrupts collective neuronal oscillations restoring a more physiological state. Overall, these results support aDFC as a better candidate for therapeutic closed-loop brain stimulation.

Data-driven control of oscillator networks with population-level measurement

Controlling complex networks of nonlinear limit-cycle oscillators is an important problem pertinent to various applications in engineering and natural sciences. While in recent years the control of oscillator populations with comprehensive biophysical models or simplified models, e.g., phase models, has seen notable advances, learning appropriate controls directly from data without prior model assumptions or pre-existing data remains a challenging and less developed area of research. In this paper, we address this problem by leveraging the network's current dynamics to iteratively learn an appropriate control online without constructing a global model of the system. We illustrate through a range of numerical simulations that the proposed technique can effectively regulate synchrony in various oscillator networks after a small number of trials using only one input and one noisy population-level output measurement. We provide a theoretical analysis of our approach, illustrate its robustness to system variations, and compare its performance with existing model-based and data-driven approaches.

Nonlinear optimal control of a mean-field model of neural population dynamics

We apply the framework of nonlinear optimal control to a biophysically realistic neural mass model, which consists of two mutually coupled populations of deterministic excitatory and inhibitory neurons. External control signals are realized by time-dependent inputs to both populations. Optimality is defined by two alternative cost functions that trade the deviation of the controlled variable from its target value against the “strength” of the control, which is quantified by the integrated 1- and 2-norms of the control signal. We focus on a bistable region in state space where one low- (“down state”) and one high-activity (“up state”) stable fixed points coexist. With methods of nonlinear optimal control, we search for the most cost-efficient control function to switch between both activity states. For a broad range of parameters, we find that cost-efficient control strategies consist of a pulse of finite duration to push the state variables only minimally into the basin of attraction of the target state. This strategy only breaks down once we impose time constraints that force the system to switch on a time scale comparable to the duration of the control pulse. Penalizing control strength via the integrated 1-norm (2-norm) yields control inputs targeting one or both populations. However, whether control inputs to the excitatory or the inhibitory population dominate, depends on the location in state space relative to the bifurcation lines. Our study highlights the applicability of nonlinear optimal control to understand neuronal processing under constraints better.

Control of coupled neural oscillations using near-periodic inputs

Deep brain stimulation (DBS) is a commonly used treatment for medication resistant Parkinson's disease and is an emerging treatment for other neurological disorders. More recently, phase-specific adaptive DBS (aDBS), whereby the application of stimulation is locked to a particular phase of tremor, has been proposed as a strategy to improve therapeutic efficacy and decrease side effects. In this work, in the context of these phase-specific aDBS strategies, we investigate the dynamical behavior of large populations of coupled neurons in response to near-periodic stimulation, namely, stimulation that is periodic except for a slowly changing amplitude and phase offset that can be used to coordinate the timing of applied input with a specified phase of model oscillations. Using an adaptive phase-amplitude reduction strategy, we illustrate that for a large population of oscillatory neurons, the temporal evolution of the associated phase distribution in response to near-periodic forcing can be captured using a reduced order model with four state variables. Subsequently, we devise and validate a closed-loop control strategy to disrupt synchronization caused by coupling. Additionally, we identify strategies for implementing the proposed control strategy in situations where underlying model equations are unavailable by estimating the necessary terms of the reduced order equations in real-time from observables.

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.

Analysis of neural clusters due to deep brain stimulation pulses

Deep brain stimulation (DBS) is an established method for treating pathological conditions such as Parkinson's disease, dystonia, Tourette syndrome, and essential tremor. While the precise mechanisms which underly the effectiveness of DBS are not fully understood, several theoretical studies of populations of neural oscillators stimulated by periodic pulses have suggested that this may be related to clustering, in which subpopulations of the neurons are synchronized, but the subpopulations are desynchronized with respect to each other. The details of the clustering behavior depend on the frequency and amplitude of the stimulation in a complicated way. In the present study, we investigate how the number of clusters and their stability properties, bifurcations, and basins of attraction can be understood in terms of one-dimensional maps defined on the circle. Moreover, we generalize this analysis to stimuli that consist of pulses with alternating properties, which provide additional degrees of freedom in the design of DBS stimuli. Our results illustrate how the complicated properties of clustering behavior for periodically forced neural oscillator populations can be understood in terms of a much simpler dynamical system.

Entrainment of a network of interacting neurons with minimum stimulating charge

Periodic pulse train stimulation is generically used to study the function of the nervous system and to counteract disease-related neuronal activity, e.g., collective periodic neuronal oscillations. The efficient control of neuronal dynamics without compromising brain tissue is key to research and clinical purposes. We here adapt the minimum charge control theory, recently developed for a single neuron, to a network of interacting neurons exhibiting collective periodic oscillations. We present a general expression for the optimal waveform, which provides an entrainment of a neural network to the stimulation frequency with a minimum absolute value of the stimulating current. As in the case of a single neuron, the optimal waveform is of bang-off-bang type, but its parameters are now determined by the parameters of the effective phase response curve of the entire network, rather than of a single neuron. The theoretical results are confirmed by three specific examples: two small-scale networks of FitzHugh-Nagumo neurons with synaptic and electric couplings, as well as a large-scale network of synaptically coupled quadratic integrate-and-fire neurons.

Optimal open-loop desynchronization of neural oscillator populations

Deep brain stimulation (DBS) is an increasingly used medical treatment for various neurological disorders. While its mechanisms are not fully understood, experimental evidence suggests that through application of periodic electrical stimulation DBS may act to desynchronize pathologically synchronized populations of neurons resulting desirable changes to a larger brain circuit. However, the underlying mathematical mechanisms by which periodic stimulation can engender desynchronization in a coupled population of neurons is not well understood. In this work, a reduced phase-amplitude reduction framework is used to characterize the desynchronizing influence of periodic stimulation on a population of coupled oscillators. Subsequently, optimal control theory allows for the design of periodic, open-loop stimuli with the capacity to destabilize completely synchronized solutions while simultaneously stabilizing rotating block solutions. This framework exploits system nonlinearities in order to strategically modify unstable Floquet exponents. In the limit of weak neural coupling, it is shown that this method only requires information about the phase response curves of the individual neurons. The effects of noise and heterogeneity are also considered and numerical results are presented. This framework could ultimately be used to inform the design of more efficient deep brain stimulation waveforms for the treatment of neurological disease.

Controlling Synchronization of Spiking Neuronal Networks by Harnessing Synaptic Plasticity

Disrupting the pathological synchronous firing patterns of neurons with high frequency stimulation is a common treatment for Parkinsonian symptoms and epileptic seizures when pharmaceutical drugs fail. In this paper, our goal is to design a desynchronization strategy for large networks of spiking neurons such that the neuronal activity of the network remains in the desynchronized regime for a long period of time after the removal of the stimulation. We develop a novel "Forced Temporal-Spike Time Stimulation (FTSTS)" strategy that harnesses the spike-timing dependent plasticity to control the synchronization of neural activity in the network by forcing the neurons in the network to artificially fire in a specific temporal pattern. Our strategy modulates the synaptic strengths of selective synapses to achieve a desired synchrony of neural activity in the network. Our simulation results show that the FTSTS strategy can effectively synchronize or desynchronize neural activity in large spiking neuron networks and keep them in the desired state for a long period of time after the removal of the external stimulation. Using simulations, we demonstrate the robustness of our strategy in desynchronizing neural activity of networks against uncertainties in the designed stimulation pulses and network parameters. Additionally, we show in simulation, how our strategy could be incorporated within the existing desynchronization strategies to improve their overall efficacy in desynchronizing large networks. Our proposed strategy provides complete control over the synchronization of neurons in large networks and can be used to either synchronize or desynchronize neural activity based on specific applications. Moreover, it can be incorporated within other desynchronization strategies to improve the efficacy of existing therapies for numerous neurological and psychiatric disorders associated with pathological synchronization.

Optimal phase control of biological oscillators using augmented phase reduction

We develop a novel optimal control algorithm to change the phase of an oscillator using a minimum energy input, which also minimizes the oscillator's transversal distance to the uncontrolled periodic orbit. Our algorithm uses a two-dimensional reduction technique based on both isochrons and isostables. We develop a novel method to eliminate cardiac alternans by connecting our control algorithm with the underlying physiological problem. We also describe how the devised algorithm can be used for spike timing control which can potentially help with motor symptoms of essential and parkinsonian tremor, and aid in treating jet lag. To demonstrate the advantages of this algorithm, we compare it with a previously proposed optimal control algorithm based on standard phase reduction for the Hopf bifurcation normal form, and models for cardiac pacemaker cells, thalamic neurons, and circadian gene regulation cycle in the suprachiasmatic nucleus. We show that our control algorithm is effective even when a large phase change is required or when the nontrivial Floquet multiplier is close to unity; in such cases, the previously proposed control algorithm fails.

Phase reduction and phase-based optimal control for biological systems: a tutorial

A powerful technique for the analysis of nonlinear oscillators is the rigorous reduction to phase models, with a single variable describing the phase of the oscillation with respect to some reference state. An analog to phase reduction has recently been proposed for systems with a stable fixed point, and phase reduction for periodic orbits has recently been extended to take into account transverse directions and higher-order terms. This tutorial gives a unified treatment of such phase reduction techniques and illustrates their use through mathematical and biological examples. It also covers the use of phase reduction for designing control algorithms which optimally change properties of the system, such as the phase of the oscillation. The control techniques are illustrated for example neural and cardiac systems.

Control Analysis and Design for Statistical Models of Spiking Networks

A popular approach to characterizing activity in neuronal networks is the use of statistical models that describe neurons in terms of their firing rates (i.e., the number of spikes produced per unit time). The output realization of a statistical model is, in essence, an n-dimensional binary time series, or pattern. While such models are commonly fit to data, they can also be postulated de novo, as a theoretical description of a given spiking network. More generally, they can model any network producing binary events as a function of time. In this paper, we rigorously develop a set of analyses that may be used to assay the controllability of a particular statistical spiking model, the point-process generalized linear model (PPGLM). Our analysis quantifies the ease or difficulty of inducing desired spiking patterns via an extrinsic input signal, thus providing a framework for basic network analysis, as well as for emerging applications such as neurostimulation design.

A new approach to optimal control of conductance-based spiking neurons

This paper presents an algorithm for solving the minimum-energy optimal control problem of conductance-based spiking neurons. The basic procedure is (1) to construct a conductance-based spiking neuron oscillator as an affine nonlinear system, (2) to formulate the optimal control problem of the affine nonlinear system as a boundary value problem based on Pontryagin's maximum principle, and (3) to solve the boundary value problem using the homotopy perturbation method. The construction of the minimum-energy optimal control in the framework of the homotopy perturbation technique is novel and valid for a broad class of nonlinear conductance-based neuron models. The applicability of our method in the FitzHugh-Nagumo and Hindmarsh-Rose models is validated by simulations.

Greater accuracy and broadened applicability of phase reduction using isostable coordinates

The applicability of phase models is generally limited by the constraint that the dynamics of a perturbed oscillator must stay near its underlying periodic orbit. Consequently, external perturbations must be sufficiently weak so that these assumptions remain valid. Using the notion of isostables of periodic orbits to provide a simplified coordinate system from which to understand the dynamics transverse to a periodic orbit, we devise a strategy to correct for changing phase dynamics for locations away from the limit cycle. Consequently, these corrected phase dynamics allow for perturbations of larger magnitude without invalidating the underlying assumptions of the reduction. The proposed reduction strategy yields a closed set of equations and can be applied to periodic orbits embedded in arbitrarily high dimensional spaces. We illustrate the utility of this strategy in two models with biological relevance. In the first application, we find that an optimal control strategy for modifying the period of oscillation can be improved with the corrected phase reduction. In the second, the corrected phase reduced dynamics are used to understand adaptation and memory effects resulting from past perturbations.

How to desynchronize quorum-sensing networks

In this paper we investigate how so-called quorum-sensing networks can be desynchronized. Such networks, which arise in many important application fields, such as systems biology, are characterized by the fact that direct communication between network nodes is superimposed to communication with a shared, environmental variable. In particular, we provide a new sufficient condition ensuring that the trajectories of these quorum-sensing networks diverge from their synchronous evolution. Then, we apply our result to study two applications.

Closed-loop deep brain stimulation by pulsatile delayed feedback with increased gap between pulse phases

Computationally it was shown that desynchronizing delayed feedback stimulation methods are effective closed-loop techniques for the control of synchronization in ensembles of interacting oscillators. We here computationally design stimulation signals for electrical stimulation of neuronal tissue that preserve the desynchronizing delayed feedback characteristics and comply with mandatory charge deposit-related safety requirements. For this, the amplitude of the high-frequency (HF) train of biphasic charge-balanced pulses used by the standard HF deep brain stimulation (DBS) is modulated by the smooth feedback signals. In this way we combine the desynchronizing delayed feedback approach with the HF DBS technique. We show that such a pulsatile delayed feedback stimulation can effectively and robustly desynchronize a network of model neurons comprising subthalamic nucleus and globus pallidus external and suggest this approach for desynchronizing closed-loop DBS. Intriguingly, an interphase gap introduced between the recharging phases of the charge-balanced biphasic pulses can significantly improve the stimulation-induced desynchronization and reduce the amount of the administered stimulation. In view of the recent experimental and clinical studies indicating a superiority of the closed-loop DBS to open-loop HF DBS, our results may contribute to a further development of effective stimulation methods for the treatment of neurological disorders characterized by abnormal neuronal synchronization.

Pulsatile desynchronizing delayed feedback for closed-loop deep brain stimulation

High-frequency (HF) deep brain stimulation (DBS) is the gold standard for the treatment of medically refractory movement disorders like Parkinson’s disease, essential tremor, and dystonia, with a significant potential for application to other neurological diseases. The standard setup of HF DBS utilizes an open-loop stimulation protocol, where a permanent HF electrical pulse train is administered to the brain target areas irrespectively of the ongoing neuronal dynamics. Recent experimental and clinical studies demonstrate that a closed-loop, adaptive DBS might be superior to the open-loop setup. We here combine the notion of the adaptive high-frequency stimulation approach, that aims at delivering stimulation adapted to the extent of appropriately detected biomarkers, with specifically desynchronizing stimulation protocols. To this end, we extend the delayed feedback stimulation methods, which are intrinsically closed-loop techniques and specifically designed to desynchronize abnormal neuronal synchronization, to pulsatile electrical brain stimulation. We show that permanent pulsatile high-frequency stimulation subjected to an amplitude modulation by linear or nonlinear delayed feedback methods can effectively and robustly desynchronize a STN-GPe network of model neurons and suggest this approach for desynchronizing closed-loop DBS.

Algorithmic design of a noise-resistant and efficient closed-loop deep brain stimulation system: A computational approach

Advances in the field of closed-loop neuromodulation call for analysis and modeling approaches capable of confronting challenges related to the complex neuronal response to stimulation and the presence of strong internal and measurement noise in neural recordings. Here we elaborate on the algorithmic aspects of a noise-resistant closed-loop subthalamic nucleus deep brain stimulation system for advanced Parkinson’s disease and treatment-refractory obsessive-compulsive disorder, ensuring remarkable performance in terms of both efficiency and selectivity of stimulation, as well as in terms of computational speed. First, we propose an efficient method drawn from dynamical systems theory, for the reliable assessment of significant nonlinear coupling between beta and high-frequency subthalamic neuronal activity, as a biomarker for feedback control. Further, we present a model-based strategy through which optimal parameters of stimulation for minimum energy desynchronizing control of neuronal activity are being identified. The strategy integrates stochastic modeling and derivative-free optimization of neural dynamics based on quadratic modeling. On the basis of numerical simulations, we demonstrate the potential of the presented modeling approach to identify, at a relatively low computational cost, stimulation settings potentially associated with a significantly higher degree of efficiency and selectivity compared with stimulation settings determined post-operatively. Our data reinforce the hypothesis that model-based control strategies are crucial for the design of novel stimulation protocols at the backstage of clinical applications.

The Geometry of Plasticity-Induced Sensitization in Isoinhibitory Rate Motifs

A well-known phenomenon in sensory perception is desensitization, wherein behavioral responses to persistent stimuli become attenuated over time. In this letter, our focus is on studying mechanisms through which desensitization may be mediated at the network level and, specifically, how sensitivity changes arise as a function of long-term plasticity. Our principal object of study is a generic isoinhibitory motif: a small excitatory-inhibitory network with recurrent inhibition. Such a motif is of interest due to its overrepresentation in laminar sensory network architectures. Here, we introduce a sensitivity analysis derived from control theory in which we characterize the fixed-energy reachable set of the motif. This set describes the regions of the phase-space that are more easily (in terms of stimulus energy) accessed, thus providing a holistic assessment of sensitivity. We specifically focus on how the geometry of this set changes due to repetitive application of a persistent stimulus. We find that for certain motif dynamics, this geometry contracts along the stimulus orientation while expanding in orthogonal directions. In other words, the motif not only desensitizes to the persistent input, but heightens its responsiveness (sensitizes) to those that are orthogonal. We develop a perturbation analysis that links this sensitization to both plasticity-induced changes in synaptic weights and the intrinsic dynamics of the network, highlighting that the effect is not purely due to weight-dependent disinhibition. Instead, this effect depends on the relative neuronal time constants and the consequent stimulus-induced drift that arises in the motif phase-space. For tightly distributed (but random) parameter ranges, sensitization is quite generic and manifests in larger recurrent E-I networks within which the motif is embedded.

Isostable reduction with applications to time-dependent partial differential equations

Isostables and isostable reduction, analogous to isochrons and phase reduction for oscillatory systems, are useful in the study of nonlinear equations which asymptotically approach a stationary solution. In this work, we present a general method for isostable reduction of partial differential equations, with the potential power to reduce the dimensionality of a nonlinear system from infinity to 1. We illustrate the utility of this reduction by applying it to two different models with biological relevance. In the first example, isostable reduction of the Fokker-Planck equation provides the necessary framework to design a simple control strategy to desynchronize a population of pathologically synchronized oscillatory neurons, as might be relevant to Parkinson's disease. Another example analyzes a nonlinear reaction-diffusion equation with relevance to action potential propagation in a cardiac system.

Suppressing epileptic activity in a neural mass model using a closed-loop proportional-integral controller

Closed-loop control is a promising deep brain stimulation (DBS) strategy that could be used to suppress high-amplitude epileptic activity. However, there are currently no analytical approaches to determine the stimulation parameters for effective and safe treatment protocols. Proportional-integral (PI) control is the most extensively used closed-loop control scheme in the field of control engineering because of its simple implementation and perfect performance. In this study, we took Jansen's neural mass model (NMM) as a test bed to develop a PI-type closed-loop controller for suppressing epileptic activity. A graphical stability analysis method was employed to determine the stabilizing region of the PI controller in the control parameter space, which provided a theoretical guideline for the choice of the PI control parameters. Furthermore, we established the relationship between the parameters of the PI controller and the parameters of the NMM in the form of a stabilizing region, which provided insights into the mechanisms that may suppress epileptic activity in the NMM. The simulation results demonstrated the validity and effectiveness of the proposed closed-loop PI control scheme.

Clustered Desynchronization from High-Frequency Deep Brain Stimulation

While high-frequency deep brain stimulation is a well established treatment for Parkinson’s disease, its underlying mechanisms remain elusive. Here, we show that two competing hypotheses, desynchronization and entrainment in a population of model neurons, may not be mutually exclusive. We find that in a noisy group of phase oscillators, high frequency perturbations can separate the population into multiple clusters, each with a nearly identical proportion of the overall population. This phenomenon can be understood by studying maps of the underlying deterministic system and is guaranteed to be observed for small noise strengths. When we apply this framework to populations of Type I and Type II neurons, we observe clustered desynchronization at many pulsing frequencies.

While high-frequency deep brain stimulation (DBS) is a decades old treatment for alleviating the motor symptoms Parkinsons disease, the way in which it alleviates these symptoms is not well understood. Making matters more complicated, some experimental results suggest that DBS works by making neurons fire more regularly, while other seemingly contradictory results suggest that DBS works by making neural firing patterns less synchronized. Here we present theoretical and numerical results with the potential to merge these competing hypotheses. For predictable DBS pulsing rates, in the presence of a small amount of noise, a population of neurons will split into distinct clusters, each containing a nearly identical proportion of the overall population. When we observe this clustering phenomenon, on a short time scale, neurons are entrained to high-frequency DBS pulsing, but on a long time scale, they desynchronize predictably.

Maximizing relaxation time in oscillator networks with implications for neurostimulation

High frequency deep brain stimulation (HF-DBS) is a pervasive clinical neurostimulation paradigm in which rapid (> 100Hz) pulses of electrical current are invasively delivered to the brain. Here, we use dynamical systems analysis to provide hypotheses regarding the frequency-specificity of the therapeutic effects of HF-DBS. Using phase oscillator-based models, we study the relaxation time of a synchronized network following impulsive stimulation. In particular, by approximating a standard DBS pulse by a finite-energy (Dirac) delta function, we show the existence of a minimum bound on the frequency of stimulation necessary to keep the network in a desynchronized regime. If, as evidence suggests, pathological synchronization is central to the pathology in DBS-responsive disorders, then the analysis gives conceptual insight into why lower frequency and/or randomized stimulation therapy is less effective, and provides a way to study alternative design strategies.

Locally optimal extracellular stimulation for chaotic desynchronization of neural populations

We use optimal control theory to design a methodology to find locally optimal stimuli for desynchronization of a model of neurons with extracellular stimulation. This methodology yields stimuli which lead to positive Lyapunov exponents, and hence desynchronizes a neural population. We analyze this methodology in the presence of interneuron coupling to make predictions about the strength of stimulation required to overcome synchronizing effects of coupling. This methodology suggests a powerful alternative to pulsatile stimuli for deep brain stimulation as it uses less energy than pulsatile stimuli, and could eliminate the time consuming tuning process.

Stochastic optimal control of single neuron spike trains

OBJECTIVE

External control of spike times in single neurons can reveal important information about a neuron's sub-threshold dynamics that lead to spiking, and has the potential to improve brain-machine interfaces and neural prostheses. The goal of this paper is the design of optimal electrical stimulation of a neuron to achieve a target spike train under the physiological constraint to not damage tissue.

APPROACH

We pose a stochastic optimal control problem to precisely specify the spike times in a leaky integrate-and-fire (LIF) model of a neuron with noise assumed to be of intrinsic or synaptic origin. In particular, we allow for the noise to be of arbitrary intensity. The optimal control problem is solved using dynamic programming when the controller has access to the voltage (closed-loop control), and using a maximum principle for the transition density when the controller only has access to the spike times (open-loop control).

MAIN RESULTS

We have developed a stochastic optimal control algorithm to obtain precise spike times. It is applicable in both the supra-threshold and sub-threshold regimes, under open-loop and closed-loop conditions and with an arbitrary noise intensity; the accuracy of control degrades with increasing intensity of the noise. Simulations show that our algorithms produce the desired results for the LIF model, but also for the case where the neuron dynamics are given by more complex models than the LIF model. This is illustrated explicitly using the Morris-Lecar spiking neuron model, for which an LIF approximation is first obtained from a spike sequence using a previously published method. We further show that a related control strategy based on the assumption that there is no noise performs poorly in comparison to our noise-based strategies. The algorithms are numerically intensive and may require efficiency refinements to achieve real-time control; in particular, the open-loop context is more numerically demanding than the closed-loop one.

SIGNIFICANCE

Our main contribution is the online feedback control of a noisy neuron through modulation of the input, taking into account physiological constraints on the control. A precise and robust targeting of neural activity based on stochastic optimal control has great potential for regulating neural activity in e.g. prosthetic applications and to improve our understanding of the basic mechanisms by which neuronal firing patterns can be controlled in vivo.

Solving Winfree's puzzle: the isochrons in the FitzHugh-Nagumo model

We consider the FitzHugh-Nagumo model, an example of a system with two time scales for which Winfree was unable to determine the overall structure of the isochrons. An isochron is the set of all points in the basin of an attracting periodic orbit that converge to this periodic orbit with the same asymptotic phase. We compute the isochrons as one-dimensional parametrised curves with a method based on the continuation of suitable two-point boundary value problems. This allows us to present in detail the geometry of how the basin of attraction is foliated by isochrons. They exhibit extreme sensitivity and feature sharp turns, which is why Winfree had difficulties finding them. We observe that the sharp turns and sensitivity of the isochrons are associated with the slow-fast nature of the FitzHugh-Nagumo system; more specifically, it occurs near its repelling (unstable) slow manifold.

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.