Optimal entrainment of neural oscillator ensembles

Optimal phase-selective entrainment of electrochemical oscillators with different phase response curves

We investigate the entrainment of electrochemical oscillators with different phase response curves (PRCs) using a global signal: the goal is to achieve the desired phase configuration using a minimum-power waveform. Establishing the desired phase relationships in a highly nonlinear networked system exhibiting significant heterogeneities, such as different conditions or parameters for the oscillators, presents a considerable challenge because different units respond differently to the common global entraining signal. In this work, we apply an optimal phase-selective entrainment technique in both a kinetic model and experiments involving electrochemical oscillators in achieving phase synchronized states. We estimate the PRCs of the oscillators at different circuit potentials and external resistance, and entrain pairs and small sets of four oscillators in various phase configurations. We show that for small PRC variations, phase assignment can be achieved using an averaged PRC in the control design. However, when the PRCs are sufficiently different, individual PRCs are needed to entrain the system with the expected phase relationships. The results show that oscillator assemblies with heterogeneous PRCs can be effectively entrained to desired phase configurations in practical settings. These findings open new avenues to applications in biological and engineered oscillator systems where synchronization patterns are essential for system performance.

Optimal external forces of the lock-in phenomena for flow past an inclined plate in uniform flow

We theoretically studied the optimal control, frequency lock-in, and phase lock-in phenomena due to the spatially localized periodic forcing in flow past an inclined plate. Although frequency lock-in is evident in many fluid phenomena, especially fluid-structure interactions, not many researchers have investigated it using a theoretical approach based on flow details. We obtained detailed information on the lock-in phenomena to external periodic forcing using phase reduction theory, a mathematical method for extracting the dynamics near the limit cycle. Furthermore, the optimal forces applied to the velocity field were determined under the condition of the minimum forcing energy and maximum lock-in range. The study of uniform periodic forces applied within spatially confined regions led to the conclusion that the effective lock-in position, which includes both the upstream and downstream areas of the plate, depends on the principal frequency of the force. The frequency lock-in range of these forces was analyzed and compared with theoretical predictions.

Engineering spatiotemporal patterns: information encoding, processing, and controllability in oscillator ensembles

The ability to finely manipulate spatiotemporal patterns displayed in neuronal populations is critical for understanding and influencing brain functions, sleep cycles, and neurological pathologies. However, such control tasks are challenged not only by the immense scale but also by the lack of real-time state measurements of neurons in the population, which deteriorates the control performance. In this paper, we formulate the control of dynamic structures in an ensemble of neuron oscillators as a tracking problem and propose a principled control technique for designing optimal stimuli that produce desired spatiotemporal patterns in a network of interacting neurons without requiring feedback information. We further reveal an interesting presentation of information encoding and processing in a neuron ensemble in terms of its controllability property. The performance of the presented technique in creating complex spatiotemporal spiking patterns is demonstrated on neural populations described by mathematically ideal and biophysical models, including the Kuramoto and Hodgkin-Huxley models, as well as real-time experiments on Wein bridge oscillators.

Optimal phase-selective entrainment of heterogeneous oscillator ensembles

We develop a framework to design optimal entrainment signals that entrain an ensemble of heterogeneous nonlinear oscillators, described by phase models, at desired phases. We explicitly take into account heterogeneity in both oscillation frequency and the type of oscillators characterized by different Phase Response Curves. The central idea is to leverage the Fourier series representation of periodic functions to decode a phase-selective entrainment task into a quadratic program. We demonstrate our approach using a variety of phase models, where we entrain the oscillators into distinct phase patterns. Also, we show how the generalizability gained from our formulation enables us to meet a wide range of design objectives and constraints, such as minimum-power, fast entrainment, and charge-balanced controls.

Developing control-theoretic objectives for large-scale brain dynamics and cognitive enhancement

The development of technologies for brain stimulation provides a means for scientists and clinicians to directly actuate the brain and nervous system. Brain stimulation has shown intriguing potential in terms of modifying particular symptom clusters in patients and behavioral characteristics of subjects. The stage is thus set for optimization of these techniques and the pursuit of more nuanced stimulation objectives, including the modification of complex cognitive functions such as memory and attention. Control theory and engineering will play a key role in the development of these methods, guiding computational and algorithmic strategies for stimulation. In particular, realizing this goal will require new development of frameworks that allow for controlling not only brain activity, but also latent dynamics that underlie neural computation and information processing. In the current opinion, we review recent progress in brain stimulation and outline challenges and potential research pathways associated with exogenous control of cognitive function.

Fast optimal entrainment of limit-cycle oscillators by strong periodic inputs via phase-amplitude reduction and Floquet theory

Optimal entrainment of limit-cycle oscillators by strong periodic inputs is studied on the basis of the phase-amplitude reduction and Floquet theory. Two methods for deriving the input waveforms that keep the system state close to the original limit cycle are proposed, which enable the use of strong inputs for entrainment. The first amplitude-feedback method uses feedback control to suppress deviations of the system state from the limit cycle, while the second amplitude-penalty method seeks an input waveform that does not excite large deviations from the limit cycle in the feedforward framework. Optimal entrainment of the van der Pol and Willamowski-Rössler oscillators with real or complex Floquet exponents is analyzed as examples. It is demonstrated that the proposed methods can achieve considerably faster entrainment and provide wider entrainment ranges than the conventional method that relies only on phase reduction.

Sparse optimization of mutual synchronization in collectively oscillating networks

We consider a pair of collectively oscillating networks of dynamical elements and optimize their internetwork coupling for efficient mutual synchronization based on the phase reduction theory developed by Nakao et al. [Chaos 28, 045103 (2018)]. The dynamical equations describing a pair of weakly coupled networks are reduced to a pair of coupled phase equations, and the linear stability of the synchronized state between the networks is represented as a function of the internetwork coupling matrix. We seek the optimal coupling by minimizing the Frobenius and L₁ norms of the internetwork coupling matrix for the prescribed linear stability of the synchronized state. Depending on the norm, either a dense or sparse internetwork coupling yielding efficient mutual synchronization of the networks is obtained. In particular, a sparse yet resilient internetwork coupling is obtained by L₁-norm optimization with additional constraints on the individual connection weights.

Semiclassical optimization of entrainment stability and phase coherence in weakly forced quantum limit-cycle oscillators

Optimal entrainment of a quantum nonlinear oscillator to a periodically modulated weak harmonic drive is studied in the semiclassical regime. By using the semiclassical phase-reduction theory recently developed for quantum nonlinear oscillators [Y. Kato, N. Yamamoto, and H. Nakao, Phys. Rev. Res. 1, 033012 (2019)10.1103/PhysRevResearch.1.033012], two types of optimization problems, one for the stability and the other for the phase coherence of the entrained state, are considered. The optimal waveforms of the periodic amplitude modulation can be derived by applying the classical optimization methods to the semiclassical phase equation that approximately describes the quantum limit-cycle dynamics. Using a quantum van der Pol oscillator with squeezing and Kerr effects as an example, the performance of optimization is numerically analyzed. It is shown that the optimized waveform for the entrainment stability yields faster entrainment to the driving signal than the case with a simple sinusoidal waveform, while that for the phase coherence yields little improvement from the sinusoidal case. These results are explained from the properties of the phase sensitivity function.

Optimization of linear and nonlinear interaction schemes for stable synchronization of weakly coupled limit-cycle oscillators

Optimization of mutual synchronization between a pair of limit-cycle oscillators with weak symmetric coupling is considered in the framework of the phase-reduction theory. By generalizing our previous study [S. Shirasaka, N. Watanabe, Y. Kawamura, and H. Nakao, Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling, Phys. Rev. E 96, 012223 (2017)2470-004510.1103/PhysRevE.96.012223] on the optimization of cross-diffusion coupling matrices between the oscillators, we consider optimization of mutual coupling signals to maximize the linear stability of the synchronized state, which are functionals of the past time sequences of the oscillator states. For the case of linear coupling, optimization of the delay time and linear filtering of coupling signals are considered. For the case of nonlinear coupling, general drive-response coupling is considered and the optimal response and driving functions are derived. The theoretical results are illustrated by numerical simulations.

Synchronization of heterogeneous oscillator populations in response to weak and strong coupling

Synchronous behavior of a population of chemical oscillators is analyzed in the presence of both weak and strong coupling. In each case, we derive upper bounds on the critical coupling strength which are valid for arbitrary populations of nonlinear, heterogeneous oscillators. For weak perturbations, infinitesimal phase response curves are used to characterize the response to coupling, and graph theoretical techniques are used to predict synchronization. In the strongly perturbed case, we observe a phase dependent perturbation threshold required to elicit an immediate spike and use this behavior for our analytical predictions. Resulting upper bounds on the critical coupling strength agree well with our experimental observations and numerical simulations. Furthermore, important system parameters which determine synchronization are different in the weak and strong coupling regimes. Our results point to new strategies by which limit cycle oscillators can be studied when the applied perturbations become strong enough to immediately reset the phase.

Inferring dynamic topology for decoding spatiotemporal structures in complex heterogeneous networks

Inferring connections forms a critical step toward understanding large and diverse complex networks. To date, reliable and efficient methods for the reconstruction of network topology from measurement data remain a challenge due to the high complexity and nonlinearity of the system dynamics. These obstacles also form a bottleneck for analyzing and controlling the dynamic structures (e.g., synchrony) and collective behavior in such complex networks. The novel contribution of this work is to develop a unified data-driven approach to reliably and efficiently reveal the dynamic topology of complex networks in different scales—from cells to societies. The developed technique provides guidelines for the refinement of experimental designs toward a comprehensive understanding of complex heterogeneous networks.

Extracting complex interactions (i.e., dynamic topologies) has been an essential, but difficult, step toward understanding large, complex, and diverse systems including biological, financial, and electrical networks. However, reliable and efficient methods for the recovery or estimation of network topology remain a challenge due to the tremendous scale of emerging systems (e.g., brain and social networks) and the inherent nonlinearity within and between individual units. We develop a unified, data-driven approach to efficiently infer connections of networks (ICON). We apply ICON to determine topology of networks of oscillators with different periodicities, degree nodes, coupling functions, and time scales, arising in silico, and in electrochemistry, neuronal networks, and groups of mice. This method enables the formulation of these large-scale, nonlinear estimation problems as a linear inverse problem that can be solved using parallel computing. Working with data from networks, ICON is robust and versatile enough to reliably reveal full and partial resonance among fast chemical oscillators, coherent circadian rhythms among hundreds of cells, and functional connectivity mediating social synchronization of circadian rhythmicity among mice over weeks.

Optimizing mutual synchronization of rhythmic spatiotemporal patterns in reaction-diffusion systems

Optimization of the stability of synchronized states between a pair of symmetrically coupled reaction-diffusion systems exhibiting rhythmic spatiotemporal patterns is studied in the framework of the phase reduction theory. The optimal linear filter that maximizes the linear stability of the in-phase synchronized state is derived for the case in which the two systems are nonlocally coupled. The optimal nonlinear interaction function that theoretically gives the largest linear stability of the in-phase synchronized state is also derived. The theory is illustrated by using typical rhythmic patterns in FitzHugh-Nagumo systems as examples.

Stability of entrainment of a continuum of coupled oscillators

Complex natural and engineered systems are ubiquitous, and their behavior is challenging to characterize and control. We examine the design of the entrainment process for an uncountably infinite collection of coupled phase oscillators that are all subject to the same periodic driving signal. In the absence of coupling, an appropriately designed input can result in each oscillator attaining the frequency of the driving signal, with a phase offset determined by its natural frequency. We consider a special case of interacting oscillators in which the coupling tends to destabilize the phase configuration to which the driving signal would send the collection in the absence of coupling. In this setting, we derive stability results that characterize the trade-off between the effects of driving and coupling, and compare these results to the well-known Kuramoto model of a collection of free-running coupled oscillators.

Optimizing stability of mutual synchronization between a pair of limit-cycle oscillators with weak cross coupling

We consider optimization of the linear stability of synchronized states between a pair of weakly coupled limit-cycle oscillators with cross coupling, where different components of state variables of the oscillators are allowed to interact. On the basis of the phase reduction theory, we derive the coupling matrix between different components of the oscillator states that maximizes the linear stability of the synchronized state under given constraints on the overall coupling intensity and the stationary phase difference. The improvement in the linear stability is illustrated by using several types of limit-cycle oscillators as examples.

Optimization of noise-induced synchronization of oscillator networks

We investigate common-noise-induced synchronization between two identical networks of coupled phase oscillators exhibiting fully locked collective oscillations. Using the collective phase description method for fully locked oscillators, we demonstrate that two noninteracting networks of coupled phase oscillators can exhibit in-phase synchronization between the networks when driven by weak common noise. We derive the Lyapunov exponent characterizing the relaxation time for synchronization and develop a method of obtaining the optimal input pattern of common noise to achieve fast synchronization. We illustrate the theory using three representative networks with heterogeneous, global, and local coupling. The theoretical results are validated by direct numerical simulations.

Phase-selective entrainment of nonlinear oscillator ensembles

The ability to organize and finely manipulate the hierarchy and timing of dynamic processes is important for understanding and influencing brain functions, sleep and metabolic cycles, and many other natural phenomena. However, establishing spatiotemporal structures in biological oscillator ensembles is a challenging task that requires controlling large collections of complex nonlinear dynamical units. In this report, we present a method to design entrainment signals that create stable phase patterns in ensembles of heterogeneous nonlinear oscillators without using state feedback information. We demonstrate the approach using experiments with electrochemical reactions on multielectrode arrays, in which we selectively assign ensemble subgroups into spatiotemporal patterns with multiple phase clusters. The experimentally confirmed mechanism elucidates the connection between the phases and natural frequencies of a collection of dynamical elements, the spatial and temporal information that is encoded within this ensemble, and how external signals can be used to retrieve this information.

Organizing and manipulating dynamic processes is important to understand and influence many natural phenomena. Here, the authors present a method to design entrainment signals that create stable phase patterns in heterogeneous nonlinear oscillators, and verify it in electrochemical reactions.

An energy-optimal approach for entrainment of uncertain circadian oscillators

We develop an approach to find an energy-optimal stimulus that entrains an ensemble of uncertain, uncoupled limit cycle oscillators. Furthermore, when entrainment occurs, the phase shift between oscillators is constrained to be less than a predetermined amount. This approach is illustrated for a model of Drosophila circadian activity, for which it performs better than a standard 24-h light-dark cycle. Because this method explicitly accounts for uncertainty in a given system and only requires information that is experimentally obtainable, it is well suited for experimental implementation and could ultimately represent what is believed to be a novel treatment for patients suffering from advanced/delayed sleep-phase syndrome.

Design of charge-balanced time-optimal stimuli for spiking neuron oscillators

In this letter, we investigate the fundamental limits on how the interspike time of a neuron oscillator can be perturbed by the application of a bounded external control input (a current stimulus) with zero net electric charge accumulation. We use phase models to study the dynamics of neurons and derive charge-balanced controls that achieve the minimum and maximum interspike times for a given bound on the control amplitude. Our derivation is valid for any arbitrary shape of the phase response curve and for any value of the given control amplitude bound. In addition, we characterize the change in the structures of the charge-balanced time-optimal controls with the allowable control amplitude. We demonstrate the applicability of the derived optimal control laws by applying them to mathematically ideal and experimentally observed neuron phase models, including the widely studied Hodgkin-Huxley phase model, and by verifying them with the corresponding original full state-space models. This work addresses a fundamental problem in the field of neural control and provides a theoretical investigation to the optimal control of oscillatory systems.

Noise-induced synchronization of oscillatory convection and its optimization

We investigate common-noise-induced phase synchronization between uncoupled identical Hele-Shaw cells exhibiting oscillatory convection. Using the phase description method for oscillatory convection, we demonstrate that the uncoupled systems of oscillatory Hele-Shaw convection can exhibit in-phase synchronization when driven by weak common noise. We derive the Lyapunov exponent determining the relaxation time for the synchronization, and develop a method for obtaining the optimal spatial pattern of the common noise to achieve synchronization. The theoretical results are confirmed by direct numerical simulations.

Optimal waveform for fast entrainment of weakly forced nonlinear oscillators

For many biological and engineered systems, a central function or design goal is to abbreviate the time required to synchronize a rhythmic process to an external forcing signal. We present a theory for deriving the input that effectively minimizes the average transient time required to entrain a phase model, which enables a practical technique for constructing fast entrainment waveforms for general nonlinear oscillators. This result is verified in numerical simulations using the Hodgkin-Huxley neuron model, and in experiments on an oscillatory electrochemical system.

Control strategies for underactuated neural ensembles driven by optogenetic stimulation

Motivated by experiments employing optogenetic stimulation of cortical regions, we consider spike control strategies for ensembles of uncoupled integrate and fire neurons with a common conductance input. We construct strategies for control of spike patterns, that is, multineuron trains of action potentials, up to some maximal spike rate determined by the neural biophysics. We emphasize a constructive role for parameter heterogeneity, and find a simple rule for controllability in pairs of neurons. In particular, we determine parameters for which common drive is not limited to inducing synchronous spiking. For large ensembles, we determine how the number of controllable neurons varies with the number of observed (recorded) neurons, and what collateral spiking occurs in the full ensemble during control of the subensemble. While complete control of spiking in every neuron is not possible with a single input, we find that a degree of subensemble control is made possible by exploiting dynamical heterogeneity. As most available technologies for neural stimulation are underactuated, in the sense that the number of target neurons far exceeds the number of independent channels of stimulation, these results suggest partial control strategies that may be important in the development of sensory neuroprosthetics and other neurocontrol applications.