We now study the perturbation of the torus of translated states from \(V_{0}\) when it is an attractor. It is well known that in this case, a torus, flow-invariant manifold persists when the equations are perturbed, as long as the perturbations are small (and smooth) enough [30].

Our aim in this section is to determine how many steady-states do actually persist when the system is perturbed by turning on long-range connections \(\epsilon_{\mathrm{LR}}\neq0\), and which phase portrait is observed on the perturbed torus.

Our method is as follows. We first analyze the remaining symmetries on the perturbed torus when \(\epsilon_{\mathrm{LR}}\neq0\). This allows one to assert the persistence of some steady-states (equilibria) which stand for points of maximal isotropy for the action of *Γ* on the perturbed torus. Moreover, when these isotropy subgroups contain the *m*-fold rotation group \(\mathbf {C}_{m}\) with \(m\geq3\), these equilibria are foci (attractive or repulsive) or nodes (the case when the Jacobian matrix has a double real eigenvalue). Now, the topology of the torus is an important constraint for the distribution of equilibria of saddle and other types. This follows from the *Poincaré–Hopf theorem*, which can be found in an abundance of literature and textbooks, and which can be stated as follows:

### Theorem 4.1

*Let*
\(\mathcal{V}\)
*be a compact orientable surface and suppose that all equilibria of a vector field defined on*
\(\mathcal{V}\)
*are non*-*degenerate* (*i*.*e*. *hyperbolic*: *all eigenvalues of Jacobian matrices have non*-*zero real part*). *Let*
*n*
*be the number of equilibria on*
\(\mathcal{V}\)
*and*
*p*
*the number of those which are of* saddle *type* (*eigenvalues have opposite signs*). *Then*
\(n-2p=\chi\), *the Euler characteristic of*
\(\mathcal{V}\).

In our case \(\mathcal{V}\) is the torus, and hence \(\chi=0\). Therefore there are an equal number of equilibria of saddle type and of non-saddle type. These two pieces informations (symmetries and Poincaré–Hopf theorem) greatly help to classify the possible phase portraits on the torus. The idea of analyzing the dynamics on a group orbit of equilibria under symmetry-breaking perturbations was introduced by [16]. It was applied in a theoretical setting to the case when the group orbit is a torus of square patterns [31] or hexagonal patterns [32] with the aim of showing the existence of robust heteroclinic cycles under certain conditions. Our aim here is different and we only focus on the cases which correspond to our model.

We next consider the square and hexagonal cases and we list the simplest possible phase portraits, that is, with the minimal number of equilibria, assuming that these are the pictures which will naturally arise in our model, unless further degeneracies are assumed. Then direct numerical simulations of the dynamics on the perturbed torus allow us to fix the actual dynamics which is induced on the invariant tori when switching on the long-range connections. Then direct numerical simulations of the dynamics on the perturbed torus allow us to select the dynamics, among the ones we predicted, which is induced on the invariant tori when switching on the long-range connections.

### On the Perturbed Torus

As the perturbed torus \(\mathcal{T}_{\epsilon_{\mathrm{LR}}}\) is diffeomorphic to the unperturbed torus \(\mathcal{T}_{0}\), we can work in the coordinates of \(\mathcal{T}_{0}\) for which the group action expression is known (see below). In the numerical experiments, the diffeomorphism is not known and thus, as we use \(\mathcal{T}_{0}\) coordinates, the projection of the patterns on this coordinate system is only approximative (see Fig. 4, for example).

To fix ideas, it is useful to be a bit more explicit. When \(\epsilon _{\mathrm{LR}}=0\), we write the unperturbed torus as

$$\mathcal{T}_{0} = \bigl\lbrace V_{0}(\cdot-\mathbf{t}), \mathbf{t}\in\varOmega\bigr\rbrace \subset \mathrm{L}^{2}_{\mathrm {per}}(\varOmega, \mathbb {R}), $$

where \(V_{0}\) is a stationary solution for \(\epsilon_{\mathrm{LR}}=0\). We assume that the torus solution is invariant by rotation and reflection which is equivalent to assuming \(V_{0}\) invariant by rotation and reflection. Under this assumption, the actions of rotations/reflections on the torus satisfy

$$ \bigl(\mathbf {R}^{\mathbf{o}},\mathbf{t}\bigr)\to \mathbf {R}^{\mathbf{o}} \cdot\mathbf{t},\quad\quad(\mathbf {K},\mathbf{t})\to \mathbf {K}\cdot\mathbf {t} . $$

(9)

This follows from \(\mathbf {R}^{\mathbf{o}}\mathbf {T}_{\mathbf{t}}\cdot V_{0} = \mathbf {T}_{\mathbf {R}^{\mathbf{o}}\mathbf{t}}V_{0}\). It implies that the action of a rotation \(\mathbf {R}^{\mathbf {a}}\) of axis **a** is given by

$$\bigl(\mathbf {R}^{\mathbf {a}},\mathbf{t}\bigr)\to \mathbf {R}^{\mathbf {a}}\cdot\mathbf{t}. $$

In our model, the lattice of translations symmetries of the torus matches the one of the PO map (see Sect. 2). Hence, \(V_{0}\) has the pinwheel periodicity:

$$ V_{0}(\mathbf {x}+2\pi k e_{1}+2\pi p e_{2})=V_{0}(\mathbf {x}), \quad\forall \mathbf {x}\in\varOmega, k,p\in\mathbb{Z}. $$

(10)

If we identify \(V_{0}(\cdot-\mathbf{t})\) and **t**, we can further simplify the study of the perturbed torus by decomposing **t** as follows:

$$\mathbf{t}=\phi_{1} e_{1}+\phi_{2} e_{2}. $$

The assumption (10) implies that \(\phi_{i}\in[0,2\pi)\).

### Remark 5

We cannot apply directly our method to the branch of stripes in Fig. 1, nor to the branch of hexagonal patterns, because the unperturbed torus generated by these patterns is not invariant by rotations.

### Square Case

Using the decomposition \(\mathbf{t}= \phi_{1} e_{1}+\phi_{2} e_{2}\), one finds:

$$ \begin{cases} \mathbf{R}^{\mathbf{o}}\cdot(\phi_{1},\phi_{2}) =(-\phi_{2},\phi_{1}),\\ \mathbf {K}\cdot(\phi_{1},\phi_{2}) =(\phi_{1},-\phi_{2}). \end{cases} $$

(11)

We collect the main results concerning the dynamics on the square in the following proposition. It is the backbone for determining the possible phase portraits on the perturbed torus.

### Proposition 4.2

*Let us assume that there is a finite number of equilibria on the perturbed torus which are all non*-*degenerate when*
\(\epsilon_{\mathrm {LR}}\neq 0\), \(\chi\geq0\). *For the lattice*
pmm, *there are at least eight equilibria on the perturbed torus*
\(\mathcal{T}_{\epsilon_{\mathrm {LR}}}\), *four of which are saddle*, *and the other four are nodes*/*foci*. *They are given by*

$$ \operatorname{Fix}\bigl(\bigl\langle \bigl( \mathbf{R}^{\mathbf{o}} \bigr)^{2}\bigr\rangle \bigr) = \bigl\lbrace (0,0), (0,\pi), (\pi,0), (\pi,\pi) \bigr\rbrace , $$

(12)

*which are centers of rotation*.

### Proof

It is easy to prove (12). Fixed point subspaces are flow invariant, this implies that \(\operatorname{Fix}(\langle(\mathbf {R}^{\mathbf{o}} )^{2}\rangle)\) is composed of stationary solutions. We also note that

$$\operatorname{Fix}\bigl(\bigl\langle \mathbf{R}^{\mathbf{o}}\bigr\rangle \bigr)= \bigl\lbrace (0,0), (\pi,\pi) \bigr\rbrace . $$

We write \(\frac{d}{dt}{\phi} = F({\phi})\) the dynamics on the torus. The equivariance implies that \(dF(\gamma\cdot\phi)\gamma=\gamma\cdot dF(\phi)\). As \({\phi}\in\operatorname{Fix}(\langle\mathbf{R}^{\mathbf{o}}\rangle)\) is *Γ*-invariant, it implies that \(dF(\phi)\) commutes with the rotation (11). Simple linear algebra shows that \(dF(\phi)\) must be a rotation matrix, i.e. that \(\operatorname{Fix}(\langle \mathbf {R}^{\mathbf{o}}\rangle)\) is composed of nodes/foci. It remains to show that this is also true for \((0,\pi)\) and \((\pi,0)\). This follows from Lemma 3.4 and

$$\mathbf {T}_{\pi(e_{1}+e_{2})}^{-1}\mathbf{R}^{\mathbf{o}}\cdot(0,\pi) = (0, \pi). $$

As \(\mathbf{R}^{\mathbf{o}}\) and \(\mathbf {T}_{\pi(e_{1}+e_{2})}\) commute, \(\mathbf {T}_{\pi(e_{1}+e_{2})}^{-1}\mathbf{R}^{\mathbf{o}}\) is of order 4, hence it is the rotation of center \((0,\pi)\). Now, we can see that the action of \(\mathbf {T}_{\pi(e_{1}+e_{2})}^{-1}\mathbf{R}^{\mathbf{o}}\) on the manifold \(\mathcal{T}_{0}\) is affine. Writing \(\gamma\equiv \mathbf {T}_{\pi (e_{1}+e_{2})}^{-1}\mathbf{R}^{\mathbf{o}}\), the equivariance gives

$$d\gamma\bigl(F(0,\pi)\bigr)\,dF(0,\pi) = dF\bigl(\gamma(0,\pi )\bigr)\,d\gamma(0, \pi). $$

From *γ* being affine and \((0,\pi)\in\operatorname{Fix}(\gamma )\), we find that \(dF(0,\pi)\) commute with \(d\gamma=d\mathbf{R}^{o}\) seen as a map on the torus. This allows one to conclude that \((0,\pi)\) is a node/focus, and also \((\pi,0)\). Being fixed points of rotations, the four nodes/foci are center of rotation symmetry.

Assume now that there are a finite number of zeros \((\phi _{i})_{i=1,\ldots,n}\) on \(\mathcal{T}_{\epsilon_{\mathrm{LR}}}\) which are all non-degenerate. Thanks to Theorem 4.1

$$\sum_{i=1}^{n}\operatorname{sign}\det dF({ \phi_{i}})=0. $$

This gives

$$-4 = \sum_{\phi_{i}\notin\operatorname{Fix} (\mathbf {R}_{s}^{2})}\operatorname{sign}\det dF({\phi_{i}}), $$

which implies the existence of at least four saddles. □

A convenient way to find the foci is to look at the fundamental domain in Fig. 2(a). These foci corresponds to the pinwheels in the fundamental domain.

We have seen that the minimal configuration, under the hypothesis that all equilibria on the perturbed torus are hyperbolic, is that of eight equilibrium points with four foci and four saddles. How does the associated phase portrait look like on the torus? The answer crucially depends upon the presence of the reflection symmetry (case when \(\theta _{0}=0\)). In this case the axes of reflection symmetry go through the equilibria. Therefore the foci are necessarily of node type: the eigenvalues of the Jacobian are double and real. The axes of reflection symmetry are invariant by the flow, which constrains the phase portrait to look like the one shown in Fig. 4. On the other hand when there is no reflection symmetry the foci are typically “true” foci: the eigenvalues of the Jacobian are complex conjugate. This allows for the possibility of periodic orbits centered at such foci, as shown in Fig. 5. These two typical situations are observed numerically, as shown on the right pictures in Figs. 4 and 5.

In order to observe limit cycles numerically, we had to change the connectivity. Indeed, if we use the prefactor \(G_{\sigma_{\theta}}=\cos \) in (7), the imaginary part of the eigenvalues of the equilibria coming from the breaking of the reflection symmetry by choosing \(\theta_{0}\notin\frac{\pi}{2}\mathbb{Z}\) is tiny: at least 3 orders of magnitude smaller than the real part. In effect, even if we break the reflection symmetry, we observe a dynamics similar to the one in Fig. 4. To have larger imaginary parts, we connect neurons with opposite preferred orientation by choosing the prefactor \(G_{\sigma_{\theta}}=\sin\) in (7). We further choose the connectivity with largest imaginary part among connectivities in Appendix B. Note that despite varying almost all parameters, we only observed the two situations as in Figs. 4 and 5 (up to a time reversal of the absence of limit cycles), as if naturally, the network equations (1) produced the simplest possibilities for all parameters that we investigated.

### Remark 6

We would like to mention that great care was taken to code the equivariance and that numerically, the error on symmetries was around 10^{−16} for the 2-norm of an arbitrary vector of 2-norm around 37. The numerical errors on the equivariance relations comes mainly from the PO map *θ* (see Eqs. (4)–(6)). Therefore, we computed the PO map by first building its fundamental domain by rotating/reflecting a basic cell and then padding this fundamental domain to cover *Ω*. This numerical accuracy of the equivariance allows one to check the predicted values of the stationary points with great accuracy using a Newton algorithm. In particular, we find numerically in Fig. 4 that the points \((\mathbf {R}^{\mathbf{o}} )^{k}\cdot(\frac{\pi}{2},0 )\), \(k=0,\ldots, 3\) are indeed saddle points.

### Hexagonal Case

We now consider the hexagonal lattice. This case is different from the square lattice because it seems from Proposition 3.6 that only counterclockwise pinwheels are center of rotations. However, it turns out that *γ* as in Lemma 3.4 is a rotation on the torus, hence yielding the other pinwheels as center of rotations.

We prove the next proposition using a different method from the one used in the proof of Proposition 4.2. Using a decomposition \(\mathbf{t}= \phi_{1} e_{1}+\phi_{2}e_{2}\), one finds

$$\mathbf {R}^{\mathbf{p}}\cdot\mathbf{t}=\left [ \begin{matrix} 2p_{1}+p_{2}-\phi_{1}-\phi_{2} \\ \phi_{1}-p_{1}+p_{2} \end{matrix} \right ],\quad\mathbf{p} = p_{1} e_{1}+p_{2}e_{2}. $$

### Proposition 4.3

*Let us assume that there is a finite number of equilibria on the perturbed torus which are all non*-*degenerate when*
\(\epsilon_{\mathrm{LR}}\neq 0\), \(\chi\geq0\). *For the lattice*
p3m1, *there are at least* 18 *equilibria on the perturbed torus*
\(\mathcal{T}_{\epsilon_{\mathrm {LR}}}\), *nine of which are saddle and the other nine are nodes*/*foci*, *given by the lattice*
\(\mathcal{L} [\frac{2\pi}{3}e_{1},\frac{2\pi }{3}e_{2} ]\)
*which are centers of rotation*. *The subgroup of translation symmetries is the lattice*
\(\mathcal{L} [\frac{2\pi }{3} (\mathbf {e}_{1}+\mathbf {e}_{2} ),\frac{2\pi}{3} (-\mathbf {e}_{1}+2\mathbf {e}_{2} ) ]\).

### Proof

In the fundamental domain, only counterclockwise pinwheels lead to a rotational symmetry. The center of rotation is then a node/focus point. In particular, we find the following node/foci points (see Sect. 3.1.2 for a definition)

$$(\phi_{1},\phi_{2})\in\biggl\{ \biggl( \frac{4\pi}{3},0 \biggr), \biggl(\frac{2\pi}{3},\frac{4\pi}{3} \biggr), \biggl(0,\frac{2\pi }{3} \biggr) \biggr\} . $$

We now look at the symmetry \(\gamma\equiv \mathbf {T}_{\frac{2\pi }{3}(e_{1}+e_{2})}^{-1}\mathbf {R}^{\mathbf{p}_{c}}\) defined in Proposition 3.6, where the axis of rotation is \(\mathbf{p}_{c} = \frac{2\pi}{3}(2e_{1}+e_{2})\). On the hexagonal torus, we find that \(\gamma=\mathbf {R}^{\mathbf{o}}\), which can be seen by writing the equations in the basis \(e_{1}\), \(e_{2}\). It yields \(\operatorname{Fix}(\gamma)= (\frac{\pi }{3},\frac{\pi}{3} )\mathbb{Z}\). Hence, these points are equilibria of node/foci type. It gives three additional nodes/foci.

We can use these centers of rotation to rotate each node/foci in order to find other equilibria. Using Lemma A.2, we have \(\gamma=\mathbf {R}^{\mathbf {v}}\cdot \mathbf {R}^{\mathbf{o}}= (\mathbf {R}^{\mathbf{u}} )^{-1}\) where \(\mathbf {v}=\frac{2\pi}{3}(e_{1}+2e_{2})\) and \(\mathbf{u}=\frac{4\pi}{3}e_{2}\). This yields the additional centers of rotations (hence node/foci):

$$(\phi_{1},\phi_{2})\in\biggl\{ \biggl(\frac{2\pi}{3},0 \biggr), \biggl(0,\frac{4\pi}{3} \biggr), \biggl(\frac{4\pi}{3}, \frac{2\pi }{3} \biggr) \biggr\} . $$

It follows that there are (at least) nine foci. Assuming that there is a finite number of zeros on \(\mathcal{T}_{\epsilon_{\mathrm{LR}}}\) which are all non-degenerate, Theorem 4.1 implies that there are as many saddles as foci.

We have shown that \(\mathcal{L} [\frac{2\pi}{3}e_{1},\frac{2\pi }{3}e_{2} ]\) is composed of centers of rotation. Using again Lemma A.1 with \(\mathbf {a}-\mathbf {b}\in\mathcal{L} [\frac {2\pi}{3}e_{1},\frac{2\pi}{3}e_{2} ]\), we find that the subgroup of translation symmetries is given by \(\mathcal{L} [\frac{2\pi }{3} (\mathbf {e}_{1}+\mathbf {e}_{2} ),\frac{2\pi}{3} (-\mathbf {e}_{1}+2\mathbf {e}_{2} ) ]\). □

We take the opportunity to show how the different nodes/foci found in Proposition 3.6 can be interpreted in cortical coordinates, as shown in Fig. 6. Briefly, we add the cortical activity as a semi-transparent overlay (transparent is when the cortical activity is high) to the PO map *θ*. We then plot small edges or hexagonal patches depending on which a subset of the hypercolumn is activated.

As in the square case, we can deduce from these results the possible phase portraits when assuming that the equilibria on the perturbed torus are all hyperbolic and that there are exactly 18 of them, nine foci and nine saddles. The situation is slightly more complicated than in the square case, but it is not difficult to show that a typical phase portrait looks like one of the diagrams shown in Fig. 7. It is numerically checked that this “minimal” situation indeed occurs.

As in the square case, we had to use \(G_{\sigma_{\theta}}=\sin\) in order to see foci with non-vanishing rotation number. Compared to the square case, we had more difficulty to keep small enough errors in the equivariance relations. This numerical error is 3 orders of magnitude bigger than for the square lattice^{Footnote 6} and given that we need to numerically solve (1) for a very long time, the errors of the symmetries seem to build up. Nevertheless, we were still able to produce simulations, corresponding to one of the possible diagrams (see Fig. 7). Except for the points \((\frac{2\pi}{3},\frac {2\pi}{3} )\mathbb{Z}\), we find the stationary points predicted by Proposition 3.6 using a Newton algorithm. Around the points \((\frac{2\pi}{3},\frac{2\pi}{3} )\mathbb{Z}\), the Newton algorithm does not converge: this seems to be caused by the presence of saddle points (also predicted in Proposition 3.6 which produces the small kinks on the trajectories around the red points (see Fig. 8)).

In the numerical simulation displayed in Fig. 8, no periodic orbit was found. The simulation seems to correspond to the predicted dynamics shown in Fig. 7, middle. On the other hand, we observe the remarkable fact that the simulations also lead to simplest possible scenarios with the smallest number of stationary points and a periodic solution.