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 the 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

Controllability analysis and design for underactuated stochastic neurocontrol

Neuroengineering has advanced tremendously over the past decade, but for sensory prosthetics and similar applications, it remains an extraordinary challenge to access neurons at the single cell resolution of most sensory encoding theories. In particular, if each neuron is “tuned” to particular stimulus features, then eliciting a target percept requires activating only neurons tuned to that percept and not others. However, most available technology is underactuated, with orders of magnitude fewer independent control inputs than neural degrees of freedom, possibly limiting its effectiveness given the inherent trade-off of resolution with network size. Here I analyze controllability for pairs of neurons receiving a common input. In particular, I extend previous work on the deterministic control problem to include stochastic membrane dynamics, treating both cases as a bifurcation problem in the noise parameter. I determine controllable regions in parameter space using a combination of mathematical analysis and numerical solution of stochastic differential and Fokker-Planck equations. I explain how boundaries between these regions change with noise level, and connect to the dynamical mechanisms by which controllability is lost. I show that in stochastic systems, in contrast to deterministic systems, expanding the allowable input space to include exponential ramps expands the parameter range over which neuron pairs are controllable. I also describe an alternative controllability definition using only mean spike times, as compared to the probability distribution of spiking within prespecified time intervals. These results could guide future control strategies in the development of sensory neuroprosthetics and other neurocontrol application

Synchronization of Coupled and Periodically Forced Chemical Oscillators

Physiological rhythms are essential in all living organisms. Such rhythms are regulated through the interactions of many cells. Deviation of a biological system from its normal rhythms can lead to physiological maladies. The tremor and symptoms associated with Parkinson\u27s disease are thought to emerge from abnormal synchrony of neuronal activity within the neural network of the brain. Deep brain stimulation is a therapeutic technique that can remove this pathological synchronization by the application of a periodic desynchronizing signal. Herein, we used the photosensitive Belousov--Zhabotinsky (BZ) chemical reaction to test the mechanism of deep brain stimulation. A collection of oscillators are initially synchronized using a regular light signal. Desynchronization is then attempted using an appropriately chosen desynchronizing signal based on information found in the phase response curve.;Coupled oscillators in various network topologies form the most common prototypical systems for studying networks of dynamical elements. In the present study, we couple discrete BZ photochemical oscillators in a network configuration. Different behaviors are observed on varying the coupling strength and the frequency heterogeneity, including incoherent oscillations to partial and full frequency entrainment. Phase clusters are organized symmetrically or non-symmetrically in phase-lag synchronization structures, a novel phase wave entrainment behavior in non-continuous media. The behavior is observed over a range of moderate coupling strengths and a broad frequency distribution of the oscillators

Optimal Control of Weakly Forced Nonlinear Oscillators

Optimal control of nonlinear oscillatory systems poses numerous theoretical and computational challenges. Motivated by applications in neuroscience, we develop tools and methods to synthesize optimal controls for nonlinear oscillators described by reduced order dynamical systems. Control of neural oscillations by external stimuli has a broad range of applications, ranging from oscillatory neurocomputers to deep brain stimulation for Parkinson\u27s disease. In this dissertation, we investigate fundamental limits on how neuron spiking behavior can be altered by the use of an external stimulus: control). Pontryagin\u27s maximum principle is employed to derive optimal controls that lead to desired spiking times of a neuron oscillator, which include minimum-power and time-optimal controls. In particular, we consider practical constraints in such optimal control designs including a bound on the control amplitude and the charge-balance constraint. The latter is important in neural stimulations used to avoid from the undesirable effects caused by accumulation of electric charge due to external stimuli. Furthermore, we extend the results in controlling a single neuron and consider a neuron ensemble. We, specifically, derive and synthesize time-optimal controls that elicit simultaneous spikes for two neuron oscillators. Robust computational methods based on homotopy perturbation techniques and pseudospectral approximations are developed and implemented to construct optimal controls for spiking and synchronizing a neuron ensemble, for which analytical solutions are intractable. We finally validate the optimal control strategies derived using the models of phase reduction by applying them to the corresponding original full state-space models. This validation is largely missing in the literature. Moreover, the derived optimal controls have been experimentally applied to control the synchronization of electrochemical oscillators. The methodology developed in this dissertation work is not limited to the control of neural oscillators and can be applied to a broad class of nonlinear oscillatory systems that have smooth dynamics

Model-Based and Machine Learning-Based Control of Biological Oscillators

Nonlinear oscillators - dynamical systems with stable periodic orbits - arise in many systems of physical, technological, and biological interest. This dissertation investigates the dynamics of such oscillators arising in biology, and develops several control algorithms to modify their collective behavior. We demonstrate that these control algorithms have potential in devising treatments for Parkinson's disease, cardiac alternans, and jet lag. Phase reduction, a classical reduction technique, has been instrumental in understanding such biological oscillators. In this dissertation, we investigate a new reduction technique called augmented phase reduction, and calculate its associated analytical expressions for six dynamically different planar systems: This helps us to understand the dynamical regimes for which the use of augmented phase reduction is advantageous over the standard phase reduction. We further this study by developing a novel optimal control algorithm based on the augmented phase reduction to change the phase of a single oscillator using a minimum energy input. We show that our control algorithm is effective even when a large phase change is required or when the nontrivial Floquet multiplier of the oscillator is close to unity; in such cases, the previously proposed control algorithm based on the standard phase reduction fails.We then devise a novel framework to control a population of biological oscillators as a whole, and change their collective behavior. Our first two control algorithms are Lyapunov-based, and our third is an optimal control algorithm which minimizes the control energy consumption while achieving the desired collective behavior of an oscillator population. We show that the developed control algorithms can synchronize, desynchronize, cluster, and phase shift the population.We continue this investigation by developing two novel machine learning control algorithms, which have a simple and intelligent structure that makes them effective even with a sparse data set. We show that these algorithms are powerful enough to control a wide variety of dynamical systems and not just biological oscillators. We conclude this study by understanding how the developed machine learning algorithms work in terms of phase reduction.In this dissertation, we have developed all these algorithms with the goal of ease of experimental implementation, for which the model parameters/training data can be measured experimentally. We close the loop on this dissertation by carrying out robustness analysis for the developed algorithms; demonstrating their resilience to noise, and thus their suitability for controlling living biological tissue. They truly hold great potential in devising treatments for Parkinson's disease, cardiac alternans, and jet lag

Extrinsic and Intrinsic Control of Integrative Processes in Neural Systems

At the simplest dynamical level, neurons can be understood as integrators. That is, neurons accumulate excitation from afferent neurons until, eventually, a threshold is reached and they produce a spike. Here, we consider the control of integrative processes in neural circuits in two contexts. First, we consider the problem of extrinsic neurocontrol, or modulating the spiking activity of neural circuits using stimulation, as is desired in a wide range of neural engineering applications. From a control-theoretic standpoint, such a problem presents several interesting nuances, including discontinuity in the dynamics due to the spiking process, and the technological limitations associated with underactuation (i.e., many neurons controlled by the same stimulation input). We consider these factors in a canonical problem of selective spiking, wherein a particular integrative neuron is controlled to a spike, while other neurons remain below threshold. This problem is solved in an optimal control framework, wherein several new geometric phenomena associated with the aforementioned nuances are revealed. Further, in an effort to enable scaling to large populations, we develop relaxations and alternative approaches, including the use of statistical models within the control design framework. Following this treatment of extrinsic control, we turn attention to a scientifically-driven question pertaining to intrinsic control, i.e., how neurons in the brain may themselves be controlling higher-level perceptual processes. We specifically postulate that neural activity is decoded, or “read-out” in terms of a drift-diffusion process, so that spiking activity drives a latent state towards a detection/perception threshold. Under this premise, we optimize the neural spiking trajectories according to several empirical cost functions and show that the optimal responses are physiologically plausible. In this vein, we also examine the nature of \u27optimal evidence\u27 for the general class of threshold-based integrative decision problems

Optimizing electrical brain stimulation for seizure disorders

University of Minnesota Ph.D. dissertation. March 2017. Major: Neuroscience. Advisor: Theoden Netoff. 1 computer file (PDF); x, 145 pages.Approximately 1% of the world population is afflicted with Epilepsy. For many patients, antiepileptic drugs do not fully control seizures. Electrical brain stimulation therapies have been effective in reducing seizure rates in some patients. While current neuromodulation devices provide a benefit to patients, efficacy can be improved by optimizing brain stimulation so that the therapy is tuned on a patient by patient basis. One optimization approach is to target deep brain regions that strongly modulate seizure prone regions. I will present data on the effects of stimulation of two different anatomical regions for seizure control, and establish my experimental platform for testing closed-loop algorithms. There are two general methods to implementing closed-loop algorithms to modulate neural activity: 1) Model-free algorithms that require a learning period to establish an optimal mapping between neural states and best therapeutic parameters, and 2) Model-based algorithms that use forward predictions of the neural system to determine the appropriate stimulation therapy to be administered. In this thesis, I will propose and test two closed-loop control schemes to control the brain activity to prevent epileptogenic activity while reducing stimulation energy. I will also present techniques to remove stimulation artifacts so that neural biomarkers can be measured while simultaneously applying stimulation. The methods I will present could potentially be implemented in next generation electrical brain stimulation hardware for seizure disorders and other neurological diseases