Designing responsive pattern generators: stable heteroclinic channel cycles for modeling and control

Non-Markovian processes on heteroclinic networks

Sets of saddle equilibria connected by trajectories are known as heteroclinic networks. Trajectories near a heteroclinic network typically spend a long period of time near one of the saddles before rapidly transitioning to the neighborhood of a different saddle. The sequence of saddles visited by a trajectory can be considered a stochastic sequence of states. In the presence of small-amplitude noise, this sequence may be either Markovian or non-Markovian, depending on the appearance of a phenomenon called lift-off at one or more saddles of the network. In this paper, we investigate how lift-off occurring at one saddle affects the dynamics near the next saddle visited, how we might determine the order of the associated Markov chain of states, and how we might calculate the transition probabilities of that Markov chain. We first review methods developed by Bakhtin to determine the map describing the dynamics near a linear saddle in the presence of noise and extend the results to include three different initial probability distributions. Using Bakhtin's map, we determine conditions under which the effect of lift-off persists as the trajectory moves past a subsequent saddle. We then propose a method for finding a lower bound for the order of this Markov chain. Many of the theoretical results in this paper are only valid in the limit of small noise, and we numerically investigate how close simulated results get to the theoretical predictions over a range of noise amplitudes and parameter values.

Stable Heteroclinic Channel Networks for Physical Human-Humanoid Robot Collaboration

Human-robot collaboration is one of the most challenging fields in robotics, as robots must understand human intentions and suitably cooperate with them in the given circumstances. But although this is one of the most investigated research areas in robotics, it is still in its infancy. In this paper, human-robot collaboration is addressed by applying a phase state system, guided by stable heteroclinic channel networks, to a humanoid robot. The base mathematical model is first defined and illustrated on a simple three-state system. Further on, an eight-state system is applied to a humanoid robot to guide it and make it perform different movements according to the forces exerted on its grippers. The movements presented in this paper are squatting, standing up, and walking forwards and backward, while the motion velocity depends on the magnitude of the applied forces. The method presented in this paper proves to be a suitable way of controlling robots by means of physical human-robot interaction. As the phase state system and the robot movements can both be further extended to make the robot execute many other tasks, the proposed method seems to provide a promising way for further investigation and realization of physical human-robot interaction.

Control for multifunctionality: bioinspired control based on feeding in Aplysia californica

Animals exhibit remarkable feats of behavioral flexibility and multifunctional control that remain challenging for robotic systems. The neural and morphological basis of multifunctionality in animals can provide a source of bioinspiration for robotic controllers. However, many existing approaches to modeling biological neural networks rely on computationally expensive models and tend to focus solely on the nervous system, often neglecting the biomechanics of the periphery. As a consequence, while these models are excellent tools for neuroscience, they fail to predict functional behavior in real time, which is a critical capability for robotic control. To meet the need for real-time multifunctional control, we have developed a hybrid Boolean model framework capable of modeling neural bursting activity and simple biomechanics at speeds faster than real time. Using this approach, we present a multifunctional model of Aplysia californica feeding that qualitatively reproduces three key feeding behaviors (biting, swallowing, and rejection), demonstrates behavioral switching in response to external sensory cues, and incorporates both known neural connectivity and a simple bioinspired mechanical model of the feeding apparatus. We demonstrate that the model can be used for formulating testable hypotheses and discuss the implications of this approach for robotic control and neuroscience.

Coupled heteroclinic networks in disguise

We consider diffusively coupled heteroclinic networks, ranging from two coupled heteroclinic cycles to small numbers of heteroclinic networks, each composed of two connected heteroclinic cycles. In these systems, we analyze patterns of synchronization as a function of the coupling strength. We find synchronized limit cycles, slowing-down states, as well as quasiperiodic motion of rotating tori solutions, transient chaos, and chaos, in general along with multistable behavior. This means that coupled heteroclinic networks easily come in disguise even when they constitute the main building blocks of the dynamics. The generated spatial patterns are rotating waves with on-site limit cycles and perturbed traveling waves from on-site quasiperiodic behavior. The bifurcation diagrams of these simple systems are in general quite intricate.

Where Computation and Dynamics Meet: Heteroclinic Network-Based Controllers in Evolutionary Robotics

In the fields of artificial neural networks and robotics, complicated, often high-dimensional systems can be designed using evolutionary/other algorithms to successfully solve very complex tasks. However, dynamical analysis of the underlying controller can often be near impossible, due to the high dimension and nonlinearities in the system. In this paper, we propose a more restricted form of controller, such that the underlying dynamical systems are forced to contain a dynamical object called a heteroclinic network. Systems containing heteroclinic networks share some properties with finite-state machines (FSMs) but are not discrete: both space and time are still described with continuous variables. Thus, we suggest that the heteroclinic networks can provide a hybrid between continuous and discrete systems. We investigate this innovated architecture in a minimal categorical perception task. The similarity of the controller to an FSM allows us to describe some of the system's behaviors as transition between states. However, other, essential behavior involves subtle ongoing interaction between the controller and the environment that eludes description at this level.

Phase descriptions of a multidimensional Ornstein-Uhlenbeck process

Stochastic oscillators play a prominent role in different fields of science. Their simplified description in terms of a phase has been advocated by different authors using distinct phase definitions in the stochastic case. One notion of phase that we put forward previously, the asymptotic phase of a stochastic oscillator, is based on the eigenfunction expansion of its probability density. More specifically, it is given by the complex argument of the eigenfunction of the backward operator corresponding to the least-negative eigenvalue. Formally, besides the "backward" phase, one can also define the "forward" phase as the complex argument of the eigenfunction of the forward Kolomogorov operator corresponding to the least-negative eigenvalue. Until now, the intuition about these phase descriptions has been limited. Here we study these definitions for a process that is analytically tractable, the two-dimensional Ornstein-Uhlenbeck process with complex eigenvalues. For this process, (i) we give explicit expressions for the two phases; (ii) we demonstrate that the isochrons are always the spokes of a wheel but that (iii) the spacing of these isochrons (their angular density) is different for backward and forward phases; (iv) we show that the isochrons of the backward phase are completely determined by the deterministic part of the vector field, whereas the forward phase also depends on the noise matrix; and (v) we demonstrate that the mean progression of the backward phase in time is always uniform, whereas this is not true for the forward phase except in the rotationally symmetric case. We illustrate our analytical results for a number of qualitatively different cases.

Heteroclinic switching between chimeras

Functional oscillator networks, such as neuronal networks in the brain, exhibit switching between metastable states involving many oscillators. We give exact results how such global dynamics can arise in paradigmatic phase oscillator networks: Higher-order network interactions give rise to metastable chimeras-localized frequency synchrony patterns-which are joined by heteroclinic connections. Moreover, we illuminate the mechanisms that underly the switching dynamics in these experimentally accessible networks.

Robustness, flexibility, and sensitivity in a multifunctional motor control model

Motor systems must adapt to perturbations and changing conditions both within and outside the body. We refer to the ability of a system to maintain performance despite perturbations as "robustness," and the ability of a system to deploy alternative strategies that improve fitness as "flexibility." Different classes of pattern-generating circuits yield dynamics with differential sensitivities to perturbations and parameter variation. Depending on the task and the type of perturbation, high sensitivity can either facilitate or hinder robustness and flexibility. Here we explore the role of multiple coexisting oscillatory modes and sensory feedback in allowing multiphasic motor pattern generation to be both robust and flexible. As a concrete example, we focus on a nominal neuromechanical model of triphasic motor patterns in the feeding apparatus of the marine mollusk Aplysia californica. We find that the model can operate within two distinct oscillatory modes and that the system exhibits bistability between the two. In the "heteroclinic mode," higher sensitivity makes the system more robust to changing mechanical loads, but less robust to internal parameter variations. In the "limit cycle mode," lower sensitivity makes the system more robust to changes in internal parameter values, but less robust to changes in mechanical load. Finally, we show that overall performance on a variable feeding task is improved when the system can flexibly transition between oscillatory modes in response to the changing demands of the task. Thus, our results suggest that the interplay of sensory feedback and multiple oscillatory modes can allow motor systems to be both robust and flexible in a variable environment.

Dynamical systems, attractors, and neural circuits

Biology is the study of dynamical systems. Yet most of us working in biology have limited pedagogical training in the theory of dynamical systems, an unfortunate historical fact that can be remedied for future generations of life scientists. In my particular field of systems neuroscience, neural circuits are rife with nonlinearities at all levels of description, rendering simple methodologies and our own intuition unreliable. Therefore, our ideas are likely to be wrong unless informed by good models. These models should be based on the mathematical theories of dynamical systems since functioning neurons are dynamic—they change their membrane potential and firing rates with time. Thus, selecting the appropriate type of dynamical system upon which to base a model is an important first step in the modeling process. This step all too easily goes awry, in part because there are many frameworks to choose from, in part because the sparsely sampled data can be consistent with a variety of dynamical processes, and in part because each modeler has a preferred modeling approach that is difficult to move away from. This brief review summarizes some of the main dynamical paradigms that can arise in neural circuits, with comments on what they can achieve computationally and what signatures might reveal their presence within empirical data. I provide examples of different dynamical systems using simple circuits of two or three cells, emphasizing that any one connectivity pattern is compatible with multiple, diverse functions.