- Open Access
Criteria for robustness of heteroclinic cycles in neural microcircuits
© Ashwin et al.; licensee Springer 2011
- Received: 6 September 2011
- Accepted: 28 November 2011
- Published: 28 November 2011
We introduce a test for robustness of heteroclinic cycles that appear in neural microcircuits modeled as coupled dynamical cells. Robust heteroclinic cycles (RHCs) can appear as robust attractors in Lotka-Volterra-type winnerless competition (WLC) models as well as in more general coupled and/or symmetric systems. It has been previously suggested that RHCs may be relevant to a range of neural activities, from encoding and binding to spatio-temporal sequence generation.
The robustness or otherwise of such cycles depends both on the coupling structure and the internal structure of the neurons. We verify that robust heteroclinic cycles can appear in systems of three identical cells, but only if we require perturbations to preserve some invariant subspaces for the individual cells. On the other hand, heteroclinic attractors can appear robustly in systems of four or more identical cells for some symmetric coupling patterns, without restriction on the internal dynamics of the cells.
- Periodic Orbit
- Invariant Subspace
- Unstable Manifold
- Cluster State
- Coupling Structure
For some time, it has been recognized that robust heteroclinic cycles (RHCs) can be attractors in dynamical systems , and that RHCs can provide useful models for the dynamics in certain biological systems. Examples include Lotka-Volterra population models  in ecology and game dynamics . Similar dynamics has been used to describe various neuronal microcircuits, in particular winnerless competition (WLC) dynamics  has been the subject of intense recent study. For example,  find conditions on the connectivity scheme of the generalised Lotka-Volterra model to guarantee the existence and structural robustness of a heteroclinic cycle in the system,  consider generalised “heteroclinic channels”,  use them as a model for sequential memory and  suggest that they may be used to describe binding problems. One question raised by these studies is whether Lotka-Volterra type dynamics is necessary to give robust heteroclinic cycles as attractors and how these cycles relate to those found in other models [9, 10]. The purpose of this paper is to show that attracting heteroclinic cycles may be robust for a variety of reasons and appear in a variety of dynamical systems that model neural microcircuits. In doing so, we give a practical test for robustness of heteroclinic cycles within any particular context and demonstrate it in practice for several examples.
This paper was motivated by a recent paper on three synaptically coupled Hodgkin-Huxley type neurons in a ring that reported robust winnerless competition between neurons  without an explicit Lotka-Volterra type structure. This manifested as a cyclic progression between states where only one neuron is active (spiking) for a period of time. During this activity, the currently active neuron inhibits the activity of the next neuron in the ring while the third neuron recovers from previous inhibition.
One of the main observations of this paper is that the coupling structure and symmetries in this system are not sufficient to guarantee robustness of the heteroclinic behaviour observed in , but robustness can be demonstrated if we consider constraints in the system. For this case it is natural to investigate the invariance of a set of affine subspaces of the system’s phase space related to the type of synaptic coupling considered. More generally, we discuss cases of heteroclinic attractors that are robust, based purely on the coupling structure and the assumption that the cells are identical.
The paper is organized as follows: In Section 2 we consider the general problem of robustness of a heteroclinic cycle. We investigate a class of dynamical systems that have affine invariant subspaces and give a necessary and sufficient condition on the dimensionality of the invariant affine subspaces for the robustness of HCs in this class of systems. We translate these conditions into appropriate conditions for coupled systems. Section 3.1 reviews a simple example of winnerless competition and demonstrates robustness for Lotka-Volterra systems, while Section 3.2 discusses the three-cell problem of Nowotny et al.. We demonstrate how the general results from Section 2 can be applied to show that the observed HC in the system (i) is not robust with respect to perturbations that only preserve its symmetry, but (ii) is robust with respect to perturbations that respect a specific set of invariant affine subspaces. Section 3.3 illustrates an example of a four-cell network of Hodgkin-Huxley type neurons where the coupling structure alone is sufficient for the robustness of HCs. We finish with a brief discussion in Section 4.
where and , the set of vector fields on with bounded global attractors.1 We say an invariant set Σ is a heteroclinic cycle (HC) if it consists of a union of hyperbolic equilibria and connecting orbits .2 We say that a heteroclinic cycle Σ is robust to perturbations in if and there is a -neighbourhood of f such that all within this neighbourhood have a heteroclinic cycle that is close to Σ.
This implies that it is not possible for Equation 2 to be satisfied for all connections. Hence our first statement is the following (which can be thought of a special case of the Kupka-Smale Theorem , see also ).
Proposition 1 A heteroclinic cycle betweenhyperbolic equilibria is never robust to generalperturbations in.
The heteroclinic cycle may however be robust to a constrained set of perturbations. We explore this in the following sections.
2.1 Conditions for robustness of heteroclinic cycles with constraints
and call the subspaces in invariant subspaces in the phase space of the dynamical systems described by .
A set of invariant affine subspaces may arise from a variety of modelling assumptions; for example,
If f is a Lotka-Volterra type population model that leaves some subspaces corresponding to the absence of one or more “species” invariant then where is the set of the invariant subspaces forced by the absence of these species.
If f is symmetric (equivariant) for some group action G and is the set of fixed point subspaces of G then because fixed point subspaces are invariant under the dynamics of equivariant systems , Theorem 1.17]. Note that for an orthogonal group action, the fixed point subspaces are linear subspaces. It is known that symmetries impose further constraints on the dynamics such as repeated eigenvalues or missing terms in Taylor expansions  but we focus here only on the invariant subspaces.
If f is a realization of a particular coupled cell system with a given coupling structure then where corresponds to the set of possible cluster states (also called synchrony subspaces or polydiagonals in the literature [15–17]).
Note that inherits a subset topology from ; for a discussion of homoclinic and heteroclinic phenomena in general and their associated bifurcations in particular, we refer to the review .
for each i. Note that there is a slight complication for the sufficient condition - it may be necessary to perturb the system slightly within to unfold the intersection to general position and remove a tangency between and . This complication has the benefit that it allows us to make statements about particular connections without needing to verify that the intersection of manifolds is transverse.
If the cycle Σ is robust to perturbations inthen Equation 8is satisfied for.
Conversely, if Equation 8is satisfied forthen there is a nearby (witharbitrarily close to f) such that Σ is a heteroclinic cycle forthat is robust to perturbations in.
Proof We will abbreviate . Because is a connection from to , there is a non-trivial intersection of within . As is the smallest invariant subspace containing , typical points will have a neighbourhood in that contain no points in any other . In a neighbourhood of this y, perturbations of f in have no restriction other than they should leave invariant.
then there will be an open dense set of perturbations of f that remove the intersection, giving lack of robustness of and hence we obtain a proof for case 1. On the other hand, if Equation 11 is not satisfied, we can choose a vector field that is identical to f except on a small neighbourhood of y - there it is chosen to preserve the connection but to perturb the manifolds so that the intersection is transverse. Transversality of the intersection then implies robustness of the connection and hence we obtain a proof for case 2. □
Just because a given heteroclinic connection is not robust due to this result does not necessarily imply that there is no robust connection from to at all. Indeed, it may be  that there are several connections from to and that perturbations will break some but not all of them. In this sense, it may be that at the same time, one heteroclinic cycle is not robust, but another heteroclinic cycle between the same equilibria is robust.
We consider robustness to perturbations that preserve the connection scheme - there are situations where a typical perturbation may break a connection but preserve a nearby connection in a larger invariant subspace. This situation will typically only occur in exceptional cases.
The structure of general robust heteroclinic cycles may be very complex even if we only examine cases forced by symmetries - they easily form networks with multiple cycles. There may be multiple or even a continuum of connections between two equilibria, and they may be embedded in more general “heteroclinic networks” where there may be connections to “heteroclinic subcycles” [16, 19, 20].
Theorem 1 does not consider any dynamical stability (attraction) properties of the heteroclinic cycles.
In what follows, we slightly abuse notation by saying that a heteroclinic cycle is robust if the cycle for an arbitrarily small perturbation of the vector field is robust.
If is not contained in then the heteroclinic cycle Σ cannot be asymptotically stable. We say that an invariant set Σ is a regular heteroclinic cycle if it consists of a union of equilibria and a set of connecting orbits with . The following result is stated in  for the case of symmetric systems.
for all i. Then the heteroclinic cycle is robust to perturbations within.
Proof Suppose that . Since is contained in by regularity of the HC, and because , we find . Hence, Equation 8 follows and we could apply Theorem 1 case 2. In fact this is a simpler case in that because the intersection must already be transverse - one does not need to consider any perturbations to force transversality of the intersection. □
2.2 Cluster states for coupled systems
is dynamically invariant for all ODEs in that class. For a given symmetry or coupling structure, we identify a list of possible cluster states and use these to test for robustness of any given heteroclinic cycle using Theorem 1.
For an open set of choices of , the heteroclinic cycle involves two saddles within the subspace and connections that are contained within in one direction and in the other. This represents a system of three identical units coupled in a specific way, where each unit has two different input types; we refer to  for details. It can be quite difficult to find a suitable function f that gives a robust heteroclinic cycle in this case. Nevertheless, once one has found a heteroclinic cycle, it can be shown to be robust using Theorem 1 (case 2).
Other examples of robust heteroclinic cycles between equilibria for systems of coupled phase oscillators are given in [22, 23]. For such systems the final state equations are obtained by reducing the dynamics to phase difference variables. In this case, each equilibrium represents the oscillatory motion of oscillators with some fixed phase difference.
2.3 Robust heteroclinic cycles between periodic orbits
In cases where a phase difference reduction is not possible, one may need to study heteroclinic cycles between periodic orbits in order to explain heteroclinic behaviour. Unlike heteroclinic cycles between equilibria, heteroclinic cycles between periodic orbits can be robust under general perturbations since for a hyperbolic periodic orbit p, . Hence, the condition Equation 2 can be satisfied. For instance, consider a system on with two hyperbolic periodic orbits p and q for which the stable and unstable manifolds , , , and are two-dimensional. In this case, and (and similarly, and ) intersect transversely, and therefore, a heteroclinic cycle between p and q can exist robustly. However, for this heteroclinic cycle only one orbit connects p to q, whereas infinitely many orbits which are backward asymptotic to p move away from the heteroclinic cycle. As a result, such a robust heteroclinic cycle cannot be asymptotically stable.
To overcome this difficulty we assume that the connections of a heteroclinic cycle between periodic orbits consist of unstable manifolds of periodic orbits and these are contained in the stable manifold of the next periodic orbit. Namely, we say an invariant set Σ is a heteroclinic cycle that contains all unstable manifolds if it consists of a union of periodic orbits and/or equilibria and a set of connecting manifolds with .
(in other words, is a sink for the dynamics reduced to) for allthen Σ is robust to perturbations within.
Proof Consider a unique orbit . Since contains a neighbourhood ofin, is robust by the same reasoning as in the proof of Theorem 2. This implies that the manifold of connections is robust for all i. □
Note that a heteroclinic cycle may contain all unstable manifolds but not be attracting even in a very weak sense (essentially asymptotically stable ). Conversely, a heteroclinic cycle may not contain all unstable manifolds but may be essentially asymptotically stable.
We discuss three examples of cases where robust heteroclinic behaviour can be found in simple neural microcircuits.
3.1 Winnerless competition in Lotka-Volterra rate models
3.2 Robustness of a heteroclinic cycle in a rate model with synaptic coupling
Numerical values of parameters used for simulation of the rate model (18); see  for a discussion of the derivation of the model and the meaning of the parameters.
2.57 × 10−3 kHz
Equilibria of (18) involved in the heteroclinic cycle for and parameters as in Table 1.
Theorem 4 There are heteroclinic cycles in the system Equation 18with parameters in Table 1. These cycles:
are not robust to perturbations that preserve thesymmetry of cyclic permutation of the cells.
are robust to perturbations that preserve the affine subspaces associated with.
Since in all cases, Theorem 1 case 1 implies that typical symmetry-preserving perturbations of the system destroy the heteroclinic cycle.
Hence the criteria of Theorem 1 (case 2) are satisfied and the heteroclinic cycle is robust with respect to -perturbations that preserve the subspaces Equation 20. □
3.3 Robustness of heteroclinic cycles for a delay-coupled Hodgkin-Huxley type model
One might suspect that Theorem 4 can be generalized to show that internal constraints might be needed to give robustness of HCs for larger numbers of cells, but this is not the case as long as the cells are assumed identical. For example [28–30] find robust cycles in systems of four or more identical, globally coupled phase oscillators with no further constraints.
To illustrate this, we give an example of a robust heteroclinic attractor for a model system of four synaptically coupled neurons. We use a modification of Rinzel’s neuron model  presented by Rubin  with synaptic coupling . Due to the global coupling of the system, the invariant subspaces are all nontrivial cluster states.
We consider the parameters , , , , , , , and synaptic coupling parameters , , .
as a measure of their phase synchronization. The neurons i and j are completely phase synchronized when .
For globally coupled networks of phase oscillators, robust heteroclinic cycles between cluster states have been found in [22, 28, 30]. Such robust heteroclinic cycles of coupled phase oscillators involve robust connections between saddle-type cluster states, where the robustness of the connections relies on them being contained within another nontrivial cluster state that corresponds to partially breaking the clusters and reforming them in a different way.
In this paper we have introduced a testable criterion for robustness for a given cycle of heteroclinic connections within constrained settings - this test involves finding the connection scheme and then applying Theorem 1. We have attempted to clarify the similarity between winnerless competition dynamics in Lotka-Volterra systems as a special case of robust heteroclinic dynamics that respect some set of invariant subspaces in a connection scheme.
Winnerless competition has previously been used to describe the competition of modes where at each mode a different neuron or neuron ensemble is active and other neurons or neuron ensembles remain inactive [8, 34]. This type of competition relies on a stable robust heteroclinic cycle where robustness is due to the constraints on the individual dynamics of neurons. However, models where constraints are only on the coupling structure can admit a general phenomenon, namely robust heteroclinic cycles between cluster states. The model analyzed in Section 3.3 is an example with RHCs between cluster states. This dynamics relies on a stable robust heteroclinic cycle where robustness is due to the invariant subspaces forced by the coupling structure. In this case, the heteroclinic cycle connects saddle equilibria or saddle periodic orbits that represent different cluster states.
We have not discussed the robustness of attraction properties of RHCs - mere existence of a RHC is not enough to guarantee that it will be an attractor, but we mention that as attraction properties are determined by open conditions on eigenvalues of the saddles (e.g. [1, 24, 35]), continuity of variation of the eigenvalues will guarantee that attractivity is also a robust property.
For larger numbers of cells in symmetric or asymmetric arrays there may be very many such invariant subspaces, giving a wide range of possible robust heteroclinic cycles. Some of these are constructed in  for small numbers of coupled cells, but up to now there does not seem to be an easy way to explore which cycles are possible and which are not within any particular system. On the other hand, verifying that a particular heteroclinic cycle is, or is not, robust is a more tractable question that we address here. Note that which cycles exist may depend not just on having a valid connection scheme for some constrained set of vector fields, but also on the constraints not preventing the existence of the appropriate saddles or connections between them.
Finally, we remark that there is evidence of metastable states in neural systems (e.g. [36–38]) that are supportive of the presence of approximate robust heteroclinic cycles. There are also suggestions that heteroclinic cycles may facilitate certain computational properties of neural systems - see for example [7, 39, 40].
1We work within the class of continuously differentiable vector fields () to ensure, by the Hartman Grobman theorem , that hyperbolic equilibria are robust - this is a minimal requirement to discuss robustness of heteroclinic cycles.
2We take the subscripts modulo p.
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