Heteroclinic channels are ensembles of trajectories that can connect saddle equilibrium points in phase space. Dynamical systems and their associated phase spaces can be used to describe natural phenomena in mathematical terms; heteroclinic channels, and the cycles (or orbits) that they produce, are features in phase space that can be designed to occupy specific locations in that space. Heteroclinic channels move trajectories from one equilibrium point to another. More formally, a heteroclinic channel is a region in phase space in which nearby trajectories are drawn closer and closer to one unique limiting trajectory, the heteroclinic orbit. Equilibria connected by heteroclinic trajectories form heteroclinic cycles and cycles can be connected to form heteroclinic networks. Heteroclinic cycles and networks naturally appear in a number of applications, such as fluid dynamics, population dynamics, and neural dynamics. In addition, dynamical systems are often used as methods for robotic control. In particular, for robotic control, the equilibrium points can correspond to robotic states, and the heteroclinic channels can provide smooth methods for switching from state to state.
Overview Heteroclinic channels (or heteroclinic orbits) are building blocks for a subset of dynamical systems that are built around connected saddle equilibrium points. Homoclinic channels/orbits join a single equilibrium point to itself, whereas heteroclinic channels join two different saddle equilibrium points in phase space. The connection is formed from the unstable manifold of the first saddle ("pushing away" from that point) to the stable manifold of the next saddle point ("pulling towards" this point). Combining at least three saddle equilibria in this way produces a heteroclinic cycle, and multiple heteroclinic cycles can be connected into heteroclinic networks.
Heteroclinic channels have both spatial and temporal features in phase space. Spatial because they affect trajectories within a certain region around themselves, and temporal because the parameters of a heteroclinic channel affect how much time a trajectory spends along that channel (or more specifically, how much time it spends around one of the saddle points). The transient nature of heteroclinic channels is important for describing their "switching" nature. That is, some neighborhood around each equilibrium point can be defined as a separate state, and the heteroclinic channel itself presents a method of switching sequentially between these states.
Heteroclinic "switching" is an important descriptor for natural phenomena, especially in neural dynamics. It has also been used as an approach for designing robotic control methods which cycle between states, whether those states are pre-defined behaviors or transient states that lead to larger behaviors.
History The mathematical image described above – a series of states with a functional mechanism for switching between them – also describes a phenomenon known as winnerless competition (WLC). Winnerless competition describes the switching phenomenon between two competitive states and was identified by Busse & Heikes in 1980 when they were investigating the change of phases in a convection cycle. However, the transient dynamics of WLC are widely agreed to first have been presented by Alfred J. Lotka, who first developed the concept to describe autocatalytic chemical reactions in 1910 and then developed an extended version in 1925 to describe ecological predator-prey relationships. In 1926, Vito Volterra independently published the same set of equations with a focus on mathematical biology, especially multi-species interactions. These equations, now known as the Lotka-Volterra equations, are widely used as a mathematical model to describe transient heteroclinic switching dynamics.
Heteroclinic cycles which describe the transition between at least three states were first described by May & Leonard in 1975. They identified a special case of the Lotka-Volterra equations for population dynamics . The re-emergence of heteroclinic cycles and the increased ability to do numerical computations as compared to the period of Lotka and Volterra, prompted a resurgence of interest in heteroclinic channels, cycles, and networks as mathematical models for transient sequential dynamics.
Heteroclinic channels have become models for neural dynamics. An example is Laurent et al. (2001) who described the neural responses of fish and insects to olfactory stimuli as a WLC system, where each stimulus and its response could be identified as a separate state within the space. The responses could be modeled in this way because of their spatial and temporal properties, which aligned with the spatiotemporal nature of WLC. Rabinovich et al. (2001) & Afraimovich et al. (2004) used WLC networks (via the Fitzhugh-Nagumo & Lotka-Volterra models, respectively) to connect the mathematical concept of stable heteroclinic channels (SHCs) to transient neural dynamics more generally, particularly other sensory processes and more abstract neural connections. Rabinovich et al. (2008) expanded this idea to larger cognitive dynamic systems, and large-scale brain networks. Stable heteroclinic channels have also been used to model neuromechanical systems. The feeding structures and associated feeding processes (stages of swallowing) of marine mollusks have been analyzed using heteroclinic channels.
Biological models have always been a source of inspiration for roboticists, especially those interested in robotic control. Since robotic control requires defining and sequencing the physical actions of the robot, models of neural dynamics can be very useful. An example of this can be found in central pattern generators, which are widely used for rhythmic robotic motion. Heteroclinic channels have been used to replicate central pattern generators for robot control. Similarly, dynamic movement primitives, another common robotic motion control system, have been adapted and made more flexible by using heteroclinic channels. In more practical applications, stable heteroclinic channels have been directly used in the control of several biologically-inspired robots
Concepts
Dynamical systems A dynamical system is a rule or set of rules that describe the evolution of a state (or a system of states) in time. The set of all possible states is called the state space. The phase space is the state space of a continuous system. Dynamical systems describe the state over time with mathematical equations, often ordinary differential equations. The current state at a particular time can be plotted as a point in phase space. The set of points over time can be plotted as a trajectory.
Stability A heteroclinic channel itself can be asymptotically stable. That is, any point near the vicinity of the channel is attracted to the heteroclinic cycle at the core of the channel. Both heteroclinic channels and cycles can be robust (or structurally stable) if, within a given parameter range, they maintain a given behavior; however, this is not required.
Stochasticity Noise is one input into a heteroclinic system to move it from one equilibrium to the next. The reason is that noise (or some other stochasticity) disturbs the system enough to move it into the vicinity of the next saddle equilibrium point in the sequence. The amount of noise required is inversely proportional to the "attractiveness" of the saddle points; the more attractive the stable part of the saddle is to the system state, the longer the trajectory will linger in its vicinity, and the more noise will be required to move the system's state off of that attractive equilibrium point. There are also other ways of moving between the equilibrium points including parametric changes, or using sensory feedback.
Control theory Control theory, in robotics, deals with the use of dynamical systems to control robotic systems. The goal of robotic control is to perform precise, coordinated actions using physical actuators in response to sensor input. Dynamical systems can be used to drive the robot to a desired state (or set of states) using sensor input to minimize actuator errors.
Mathematical definition An equilibrium point in a dynamical system is a solution to the system of differential equations describing a trajectory that does not change with time. Equilibrium points can be described by their stability, which are often determined by the eigenvalues of the system's Jacobian matrix. In general, the eigenvalues of a saddle point have non-zero real parts, at least one of the real parts is positive and at least one of the real parts is negative. Any eigenvalue with a negative real value indicates a stable manifold of the saddle which attracts trajectories, whereas any eigenvalue with a positive real value indicates the unstable manifold of the saddle which repels trajectories.
Phase space definition (from Heteroclinic orbit) Let be the ordinary differential equation describing a continuous dynamical system. If there are equilibria at and , then a solution is a heteroclinic connection from to if
as
and
as
This implies that the connection is contained in the stable manifold of and the unstable manifold of .
Neural dynamics examples Neural dynamics are the non-linear dynamics that describe neural processes, from single neurons to cognitive processes and large-scale neural systems.
Lotka-Volterra model This model was first presented independently by Alfred J. Lotka for autocatalytic chemical reactions and then again for biological species in competition by Vito Volterra from a mathematical biology perspective. Originally, this model was only considered for two species: the two chemical species in the reaction, or a predator-prey situation in a shared environment.
The original equations were based on the logistic population equation, which is popularly used in ecology.
where is the size or concentration of a species at a given time, is the growth rate and is the carrying capacity of that species.
Lotka incorporated a term for the interaction between species and, with some generalization, the series of equations can be written as follows:
In this definition, is the size or concentration of the -th species and is the total number of species. The interaction between each species is described by the matrix .
May-Leonard expansion May and Leonard expanded the Lotka-Volterra equations by investigating the system in which three species interact with each other (i.e., ). They found that for a system in which each equilibrium point is a saddle with an dimensional stable manifold, and the unstable manifold connects the points sequentially, the equation above can be re-written as follows:
Explicitly for , this becomes
where the coupling matrix, , is given by
In this model, the stability of the saddle equilibria can be easily determined. The stability requirements for the formation of a stable heteroclinic cycle are with either or .
It was noted in this work that the system never asymptotically reaches any of the equilibrium points, but the amount of time the trajectory spends at each equilibrium point increases with time. In ecological terms, this suggests that a single population would eventually "beat out" the other two. May & Leonard noted that this is not a practical result in biology (and also see).
Winnerless Competition framework