Criteria for robustness of heteroclinic cycles in neural microcircuits

2.1 Conditions for robustness of heteroclinic cycles with constraints

and call the subspaces in $\mathcal{I}$invariant subspaces in the phase space of the dynamical systems described by $f\in {\mathcal{X}}_{\mathcal{I}}$.

A set of invariant affine subspaces $\mathcal{I}$ may arise from a variety of modelling assumptions; for example,

Note that ${\mathcal{X}}_{\mathcal{I}}$ inherits a subset topology from $\mathcal{X}$; for a discussion of homoclinic and heteroclinic phenomena in general and their associated bifurcations in particular, we refer to the review [13].

for each i. Note that there is a slight complication for the sufficient condition - it may be necessary to perturb the system slightly within ${\mathcal{X}}_{\mathcal{I}}$ to unfold the intersection to general position and remove a tangency between $W^u(x_i)$ and $W^s(x_{i+1})$. 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.

Proof We will abbreviate $I_c:=I_{c(i)}$. Because $s_i$ is a connection from $x_i$ to $x_{i+1}$, there is a non-trivial intersection of $W^u(x_i)\cap W^s(x_{i+1})$ within $I_c$. As $I_c$ is the smallest invariant subspace containing $s_i$, typical points $y\in s_i$ will have a neighbourhood in $I_c$ that contain no points in any other $I_j$. In a neighbourhood of this y, perturbations of f in ${\mathcal{X}}_{\mathcal{I}}$ have no restriction other than they should leave $I_c$ invariant.

then there will be an open dense set of perturbations of f that remove the intersection, giving lack of robustness of $s_i$ 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 $\tilde{f}$ 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. □

If $W^u(x_i)$ is not contained in $W^s(x_{i+1})$ 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 $s_i\subset W^u(x_i)$ with $W^u(x_i)\subset W^s(x_{i+1})$. The following result is stated in [13] for the case of symmetric systems.

for all i. Then the heteroclinic cycle is robust to perturbations within${\mathcal{X}}_{\mathcal{I}}$.

Proof Suppose that $W^s(x_{i+1})\supset I_{c(i)}$. Since $W^u(x_i)$ is contained in $W^s(x_{i+1})$ by regularity of the HC, and because $I_{c(i)}\supseteq s_i=W^u(x_i)$, we find $dim(W^u(x_i)\cap I_{c(i)})+dim(W^s(x_{i+1})\cap I_{c(i)})=dim(W^u(x_i))+dim(I_{c(i)})\ge dim(I_{c(i)})+1$. Hence, Equation 8 follows and we could apply Theorem 1 case 2. In fact this is a simpler case in that because $dim(W^s(x_{i+1})\cap I_{c(i)})=dim(I_{c(i)})$ 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 $f(x,y,z)$, the heteroclinic cycle involves two saddles within the subspace $I_1:=\{x=y=z\}$ and connections that are contained within $I_2:=\{x=y\}$ in one direction and $I_3:=\{x=z\}$ 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 [15] 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, $dim(W^u(p))+dim(W^s(p))=n+1$. Hence, the condition Equation 2 can be satisfied. For instance, consider a system on ${\mathbb{R}}^3$ with two hyperbolic periodic orbits p and q for which the stable and unstable manifolds $W^s(p)$, $W^u(p)$, $W^s(q)$, and $W^u(q)$ are two-dimensional. In this case, $W^u(p)$ and $W^s(q)$ (and similarly, $W^u(q)$ and $W^s(p)$) 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 $\{x_i:i=1,\dots ,p\}$ and a set of connecting manifolds $S_i=W^u(x_i)$ with $W^u(x_i)\subset W^s(x_{i+1})$.

(in other words, $x_{i+1}$is a sink for the dynamics reduced to$I_{c(i)}$) for all$i=1,\dots ,p$then Σ is robust to perturbations within${\mathcal{X}}_{\mathcal{I}}$.

Proof Consider a unique orbit $s_i\subset S_i$. Since $W^s(x_{i+1})$contains a neighbourhood of$x_{i+1}$in$I_{c(i)}$, $s_i$ is robust by the same reasoning as in the proof of Theorem 2. This implies that the manifold of connections $S_i$ 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 [24]). Conversely, a heteroclinic cycle may not contain all unstable manifolds but may be essentially asymptotically stable.

3.1 Winnerless competition in Lotka-Volterra rate models

3.2 Robustness of a heteroclinic cycle in a rate model with synaptic coupling

A typical timeseries showing an attracting heteroclinic cycle for this system is shown in Figure 1.

Theorem 4 There are heteroclinic cycles in the system Equation 18with parameters in Table 1. These cycles:

Since $c(i)=1$ 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 $C^1$-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 [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.

We consider the parameters $v_{\mathrm{Na}}=50$, $v_{\mathrm{K}}=-77$, $v_{\mathrm{L}}=-54.4$, $g_{\mathrm{Na}}=120$, $g_{\mathrm{K}}=36$, $g_{\mathrm{L}}=0.3$, $c=1$, $I=10$ and synaptic coupling parameters $g_{\mathrm{syn}}=0.08$, $v_{\mathrm{syn}}=0$, ${\tau }_{\mathrm{syn}}=20$.

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 $h-s$ plane (see Figure 2). A reference point $(h_{\mathrm{ref}},s_{\mathrm{ref}})=(0.2,0.85)$ is chosen and the approximate phase is given as

$\theta =arctan\left[\frac{s-s_{\mathrm{ref}}}{h-h_{\mathrm{ref}}}\right].$

as a measure of their phase synchronization. The neurons i and j are completely phase synchronized when ${\rho }_{ij}=1$.

and $x_1,x_2\in I_3:=I_1\cap I_2$. Theorem 3 implies that as long as (a) the periodic orbits $x_1$ (resp. $x_2$) are hyperbolic, and (b) they are sinks when considered within the subspaces $I_1$ (resp. $I_2$) then the connection is robust. Condition (a) is generically satisfied. We have checked (b) using simulations by choosing different initial conditions for the dynamics reduced to $I_1$ and $I_2$.

Coupled phase oscillators are used as simplified models for weakly coupled limit cycle oscillators, and one can find one-to-one 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.

For $N=5$, more complex heteroclinic cycles can appear as seen in Figure 5. This is a heteroclinic cycle that connects different cluster states of type$(2,2,1)$. Note that the transition times between clusters are fixed but the duration of stay at each cluster gets longer and longer - a feature of attracting heteroclinic cycles. In the case of noisy systems, the dynamics switches from one cluster to another randomly around a graph of connections between symmetric cluster states [10].

For globally coupled networks of $N\ge 4$ 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.