Abstract
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 LotkaVolterratype 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 spatiotemporal 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.
1 Introduction
For some time, it has been recognized that robust heteroclinic cycles (RHCs) can be attractors in dynamical systems [1], and that RHCs can provide useful models for the dynamics in certain biological systems. Examples include LotkaVolterra population models [2] in ecology and game dynamics [3]. Similar dynamics has been used to describe various neuronal microcircuits, in particular winnerless competition (WLC) dynamics [4] has been the subject of intense recent study. For example, [5] find conditions on the connectivity scheme of the generalised LotkaVolterra model to guarantee the existence and structural robustness of a heteroclinic cycle in the system, [6] consider generalised “heteroclinic channels”, [7] use them as a model for sequential memory and [8] suggest that they may be used to describe binding problems. One question raised by these studies is whether LotkaVolterra 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 HodgkinHuxley type neurons in a ring that reported robust winnerless competition between neurons [11] without an explicit LotkaVolterra 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 [11], 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 LotkaVolterra systems, while Section 3.2 discusses the threecell
problem of Nowotny et al.[11]. 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
2 Robustness of heteroclinic cycles
Suppose we have a dynamical system given by a system of first order differential equations
where
Let us suppose that
Using the fact that
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 KupkaSmale Theorem [12], see also [13]).
Proposition 1A heteroclinic cycle between
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
A subset
that are closed under intersection; i.e. the intersection
and call the subspaces in
A set of invariant affine subspaces
• If f is a LotkaVolterra type population model that leaves some subspaces corresponding
to the absence of one or more “species” invariant then
• If f is symmetric (equivariant) for some group action G and
• If f is a realization of a particular coupled cell system with a given coupling structure
then
Note that
Suppose that for a vector field
i.e. the smallest subspace in
The following theorem gives necessary and sufficient conditions for such a heteroclinic
cycle to be robust to perturbations in
for each i. Note that there is a slight complication for the sufficient condition  it may be
necessary to perturb the system slightly within
Theorem 1Let Σ be a heteroclinic cycle for
1. If the cycle Σ is robust to perturbations in
2. Conversely, if Equation 8is satisfied for
Proof We will abbreviate
The stability of the intersection of the unstable and stable manifolds depends on
the dimension of the unstable manifolds (also called the Morse index[13]) for these equilibria for the vector field restricted to
and within P, the invariant manifolds have dimensions
The intersection of these invariant manifolds may not be transverse within P, but it will be for a dense set of nearby vector fields. In particular, if
then there will be an open dense set of perturbations of f that remove the intersection, giving lack of robustness of
Note that caution is necessary in interpreting this result for a number of reasons:
1. Just because a given heteroclinic connection is not robust due to this result does not necessarily imply
that there is no robust connection from
2. 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.
3. 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].
4. Theorem 1 does not consider any dynamical stability (attraction) properties of the heteroclinic cycles.
5. 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
Theorem 2Suppose that Σ is a regular heteroclinic cycle for
for alli. Then the heteroclinic cycle is robust to perturbations within
Proof Suppose that
2.2 Cluster states for coupled systems
RHCs may appear in coupled systems due to a variety of constraints from the coupling
structure  these are associated with cluster states (also called synchrony subspaces
[15] or polydiagonals for the network [21]). Consider a network of N systems each with phase space
for
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.
We remark that the simplest (and indeed only, up to relabelling) coupling structure for a network of three identical cells found by [15] to admit heteroclinic cycles can be represented as a system of the form
For an open set of choices of
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,
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
Theorem 3Suppose that Σ is a heteroclinic cycle that contains all unstable manifolds for
(in other words,
Proof Consider a unique orbit
Note that a heteroclinic cycle may contain all unstable manifolds but not be attracting even in a very weak sense (essentially asymptotically stable [24]). Conversely, a heteroclinic cycle may not contain all unstable manifolds but may be essentially asymptotically stable.
3 Robust heteroclinic behaviour in neural models
We discuss three examples of cases where robust heteroclinic behaviour can be found in simple neural microcircuits.
3.1 Winnerless competition in LotkaVolterra rate models
The review [25] includes a discussion of winnerless competition and related phenomena. This has focused on the dynamics of LotkaVolterra type models for firing rates, justified by an approximation of Fukai and Tanaka [26]. In their most general form, these are written as
where
for example
with
and
Finally, from Theorem 1 case 2, we can conclude that the heteroclinic cycle between
saddle equilibria
3.2 Robustness of a heteroclinic cycle in a rate model with synaptic coupling
We now turn to the robustness of heteroclinic cycles in a specific model of
where time variable t is in ms. The unitless dynamical variables
characterises the rate response of the neurons to input current. We have introduced
a smoothing factor
A typical timeseries showing an attracting heteroclinic cycle for this system is shown in Figure 1.
Fig. 1. A trajectory approaching a heteroclinic cycle for the rate model (18) with
The heteroclinic cycle
Theorem 4There are heteroclinic cycles in the system Equation 18with parameters in Table 1. These cycles:
• are not robust to perturbations that preserve the
• are robust to perturbations that preserve the affine subspaces associated with
Proof(We do not rigorously prove that the heteroclinic cycles exists; this should in principle
be possible via rigorous methods with an error bounded integrator  see for example[27].) To show the first part, note that the only invariant subspaces in
Since
To see the second part, let us consider the set of vector fields on
Examining the equilibria in Table 2 we note that the
For the particular choice of parameters in Table 1, there is a heteroclinic cycle between three equilibria
Hence the criteria of Theorem 1 (case 2) are satisfied and the heteroclinic cycle
is robust with respect to
3.3 Robustness of heteroclinic cycles for a delaycoupled HodgkinHuxley 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 [2830] 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 [31] presented by Rubin [32] with synaptic coupling [32]. Due to the global coupling of the system, the invariant subspaces are all nontrivial cluster states.
Consider N alltoall synaptically coupled neurons with delay coupling (using units of mV for voltages, ms for time, mS/cm^{2} for conductances, μA/cm^{2} for currents, and μF/cm^{2} for capacitance):
for
We consider the parameters
The dynamics of this model is oscillatory for these parameter values. For the purpose
of visualizing the dynamics, we define an approximate phase using a projection of
the oscillation signal onto the
Fig. 2. Oscillation of an uncoupled HodgkinHuxley type neuron (21) in the
For two different neurons we use the synchronization index
as a measure of their phase synchronization. The neurons i and j are completely phase synchronized when
For
where
and
Fig. 3. A solution of (21) for
Coupled phase oscillators are used as simplified models for weakly coupled limit cycle oscillators, and one can find onetoone correspondence between solutions if the coupling is weak enough [33]. In particular, the heteroclinic cycle depicted in Figure 3 corresponds in clustering type to a heteroclinic cycle found in [22] (see Figure 4) for a system of four globally coupled phase oscillators. The two saddle periodic orbits mentioned above correspond to the two saddle equilibria in Figure 4.
Fig. 4. A robust heteroclinic cycle for four alltoall coupled phase oscillator system analogous
to the cycle found in Figure 3 for the HodgkinHuxley type system. The heteroclinic cycle consists of two saddle
equilibria
For
Fig. 5. A trajectory of the system of
For globally coupled networks of
4 Discussion
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 LotkaVolterra 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 [15] 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. [3638]) 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].
Competing interests
The authors declare that they have no competing interests.
Footnotes
^{1}We work within the class of continuously differentiable vector fields (
^{2}We take the subscripts modulo p.
References

Krupa, M: Robust heteroclinic cycles. J. Nonlinear Sci.. 7(2), 129–176 (1997). Publisher Full Text

May, RM, Leonard, WJ: Nonlinear aspects of competition between three species. SIAM J. Appl. Math.. 29(2), 243–253 (1975). Publisher Full Text

Hofbauer, J, Sigmund, K: Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge (1998)

Rabinovich, MI, Volkovskii, A, Lecanda, P, Huerta, R, Abarbanel, HDI, Laurent, G: Dynamical encoding by networks of competing neuron groups: Winnerless competition. Phys. Rev. Lett.. 87(6), (2001)

Afraimovich, VS, Rabinovich, MI, Varona, P: Heteroclinic contours in neural ensembles and the winnerless competition principle. Int. J. Bifurcat. Chaos. 14(4), 1195–1208 (2004). Publisher Full Text

Bick, C, Rabinovich, MI: On the occurrence of stable heteroclinic channels in LotkaVolterra models. Dyn. Syst.. 25, 97–110 (2010). Publisher Full Text

Seliger, P, Tsimring, LS, Rabinovich, MI: Dynamicsbased sequential memory: winnerless competition of patterns. Phys. Rev. E (3). 67, (2003)

Rabinovich, MI, Afraimovich, VS, Varona, P: Heteroclinic binding. Dynamical Systems. 25(3), 433–442 (2010). Publisher Full Text

Wordsworth, J, Ashwin, P: Spatiotemporal coding of inputs for a system of globally coupled phase oscillators. Phys. Rev. E. 78, (2008)

Ashwin, P, Orosz, G, Wordsworth, J, Townley, S: Dynamics on networks of clustered states for globally coupled phase oscillators. SIAM J. Appl. Dyn. Syst.. 6(4), 728–758 (2007). Publisher Full Text

Nowotny, T, Rabinovich, MI: Dynamical origin of independent spiking and bursting activity in neural microcircuits. Phys. Rev. Lett.. 98, (2007)

Palis, J, de Melo, W: Geometric Theory of Dynamical Systems, SpringerVerlag, New York (1982)

Homburg, AJ, Sandstede, B: Homoclinic and heteroclinic bifurcations in vector fields. Handbook of Dynamical Systems III. 379–524 (2010)

Golubitsky, M, Stewart, I: The Symmetry Perspective, Birkhäuser Verlag, Basel (2003)

Aguiar, MAD, Ashwin, P, Dias, APS, Field, M: Dynamics of coupled cell networks: synchrony, heteroclinic cycles and inflation. J. Nonlinear Sci.. 21, 271–323 (2011). Publisher Full Text

Field, M: Combinatorial dynamics. Dynamical Systems. 19, 217–243 (2004). Publisher Full Text

Golubitsky, M, Stewart, I: Nonlinear dynamics of networks: the groupoid formalism. Bull. Am. Math. Soc.. 43, 305–364 (2006). Publisher Full Text

Field, M: Lectures on Bifurcations, Dynamics and Symmetry, Longman, Harlow (1996)

Ashwin, P, Field, M: Heteroclinic networks in coupled cell systems. Arch. Ration. Mech. Anal.. 148(2), 107–143 (1999). Publisher Full Text

Chawanya, T, Ashwin, P: A minimal system with a depthtwo heteroclinic network. Dynamical Systems. 25, 397–412 (2010). Publisher Full Text

Stewart, I, Golubitsky, M, Pivato, M: Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM J. Appl. Dyn. Sys.. 2(4), 609–646 (2003). Publisher Full Text

Ashwin, P, Burylko, O, Maistrenko, Y: Bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators. Physica D. 237, 454–466 (2008). Publisher Full Text

Karabacak, O, Ashwin, P: Heteroclinic ratchets in networks of coupled oscillators. J. Nonlinear Sci.. 20, 105–129 (2010). Publisher Full Text

Podvigina, O, Ashwin, P: On local attraction properties and a stability index for heteroclinic connections. Nonlinearity. 24(3), 887–929 (2011). Publisher Full Text

Rabinovich, MI, Varona, P, Selverston, AI, Abarbanel, HDI: Dynamical principles in neuroscience. Rev. Mod. Phys.. 95, 519–536 (2006)

Fukai, T, Tanaka, S: A simple neural network exhibiting selective activation of neuronal ensembles: from winnertakeall to winnersshareall. Neural Comput.. 9, 77–97 (1997). PubMed Abstract  Publisher Full Text

Proceedings of the International Conference on Interval Methods and Computer Aided Proofs in Science and Engineering  Interval’96. Springer, Dordrecht (1997) [Held in Würzburg, September 30October 2, 1996, edited by J. Wolff von Gudenberg, Reliab. Comput. 3(3) (1997)]

Ashwin, P, Swift, JW: The dynamics of n weakly coupled identical oscillators. J. Nonlinear Sci.. 2, 69–108 (1992). Publisher Full Text

Ashwin, P, Burylko, O, Maistrenko, Y, Popovych, O: Extreme sensitivity to detuning for globally coupled phase oscillators. Phys. Rev. Lett.. 96(5), (2006)

Hansel, D, Mato, G, Meunier, C: Clustering and slow switching in globally coupled phase oscillators. Phys. Rev. E. 48(5), 3470–3477 (1993). Publisher Full Text

Rinzel, J: Excitation dynamics: insights from simplified membrane models. Fed. Proc.. 44, 2944–2946 (1985). PubMed Abstract

Rubin, JI: Surprising effects of synaptic excitation. J. Comput. Neurosci.. 18(3), 333–342 (2005). PubMed Abstract  Publisher Full Text

Hansel, D, Mato, G, Meunier, C: Phase dynamics for weakly coupled HodgkinHuxley neurons. Europhys. Lett.. 23, 367–372 (1993). Publisher Full Text

Komarov, MA, Osipov, GV, Suykens, JAK: Sequentially activated groups in neural networks. Europhys. Lett.. 86(6), (2009)

Krupa, M, Melbourne, I: Asymptotic stability of heteroclinic cycles in systems with symmetry. Ergod. Theory Dyn. Syst.. 15, 121–147 (1995)

Beggs, J, Plenz, D: Neuronal avalanches are diverse and precise activity patterns that are stable for many hours in cortical slice cultures. J. Neurosci.. 24, 5216–5229 (2004). PubMed Abstract  Publisher Full Text

Mazor, O, Laurent, G: Transient dynamics versus fixed points in odor representations by locust antennal lobe projection neurons. Neuron. 48, 661–673 (2005). PubMed Abstract  Publisher Full Text

PerezOrive, J, Mazor, O, Turner, GC, Cassenaer, S, Wilson, RI, Laurent, G: Oscillations and sparsening of odor representations in the mushroom body. Science. 297, 359–365 (2002). PubMed Abstract  Publisher Full Text

Ashwin, P, Borresen, J: Encoding via conjugate symmetries of slow oscillations for globally coupled oscillators. Phys. Rev. E. 70(2), (2004)

Neves, FS, Timme, M: Controlled perturbationinduced switching in pulsecoupled oscillator networks. J. Phys. A. 42, (2009)