Bifurcations on the ring

ee value:

the value the trace crosses 0 for a whole wavenumber, in this case 5; the uncorrected a∗_eevalue, the value for which the determinant exactly equals 0 at a possibly non-integer wavenumber, was ≈7.3741). Thus, we expect the stimulus-free network to have a stable low steady state (at (u, v) = (0, 0)) that will lose stability to spatiotemporal oscillations with a wavenumber of 5 at the above a∗_ee value.

By following the low steady state in AUTO, we find it loses stability at a Hopf bifurcation at a∗_ee≈ 7.3746 (Fig17A). In fact, for this set of parameters the analytically and numerically computed values differ by less than 1 × 10−10. We note that since the homogeneous network loses stability to nonzero modes, this bifurcation is known as a Turing-Hopf bifurcation. Just above this value (in particular, e.g., at aee= 7.367), we find a variety of stable spatiotemporal

patterns of different wave numbers to which the network evolves by beginning with either random initial conditions or initial conditions given by cosines (e.g., 0.1 cos 2πlj/N for uj and

vj, j ∈ (1, ..., N )). Note, we use l to differentiate from both the stimulus wavenumber k and

the network wavenumber m. While we tried all of the permissible wavenumbers l ∈ P (see Sec 3.2.2) with a range of amplitudes, the network only evolved to spatiotemporal patterns with wavenumbers m = 3, 4, 5, and a sort of wave-like pattern involving a combination of wavenumbers 2 and 3, described below. These are shown in Fig 17A. Note that the wavenumber of the network, m, is measured by counting the number of high-activity (or low-activity) regions there at any given time slice (horizontal slices in the images in Fig17A). Since the low steady state is lost to oscillatory activity at non-zero wavenumbers m, including m∗ = 5, this Hopf bifurcation corresponds to a TH bifurcation, as expected from the system linearization.

There are several patterns that occur with aee just above a∗ee that involve the critical

mode, m∗ = 5. First, we find two highly-structured stable low-amplitude patterns – (i ) and (ii ) in Fig 18A. (i ) looks like a checker board, with alternating rectangular regions of high and low activity with central regions exhibiting the highest (or lowest) activity. (ii ) is a wave pattern with 5 (high-activity) stripes. We note the obvious similarities to the two spontaneous patterns the network on the torus exhibits; however, we caution that even

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q u₁ A) a_ee* k=0 k=10 k=12 _k=5 k=0 k=6 k=5 k=10 Stable steady state

Unstable steady state Hopf bifurcation

B) C)

Time Space

-1e-3

(i) (ii) (iii)

(iv) (vi) 1e-3 0.8 -0.1 7e-3 -6e-3 -0.1 (v) 0.8 7e-3 -6e-3 -0.1 0.8 -0.1 0.8 0ms 125ms 200ms 0ms 7.37 ~ ~

Figure 17: Bifurcations on the ring. The network, extended in one spatial dimension and with periodic boundary conditions imposed, shows similar behaviors as for the network on the torus. Using XPP-AUTO, we are able to precisely identify bifurcations. (A) Above a critical value aee = a∗ee ≈ 7.37, the low steady state of the stimulus-free network is lost

to spatiotemporal patterns. The red line to the left of a∗_ee indicates the steady state is stable, while the black line to the right of a∗_ee indicates the steady state is unstable. Various patterns found for aee ' a∗ee are shown. Three patterns at the critical wavenumber 5 are

found (where we determine the wavenumber by counting the number of high or low regions of activity for a given time, corresponding to horizontal slices in the figures shown): (i ) small-amplitude alternating checkerboard oscillations, (ii ) small-amplitude waves, and (iii ) large-amplitude alternating oscillations. The other oscillatory solutions found were: (iv ) staggered oscillations that alternated between a wavenumber of 2 and 3, and alternating oscillations with (v ) wavenumber 3, and (vi ) wavenumber 4. (B) Using a simple cosine stimulus of wavenumber k, the steady states of the network are lost at Hopf bifurcations for different values of q(k). (C) Different spatiotemporal patterns can form, depending on the values of k, q. We see that the stimulus wavenumber, k, need not match the wavenumber of the network pattern that forms, m. For constant stimuli (k = 0), either small-amplitude checkerboard oscillations of wavenumber m = 5 or more fluid, wavelike large-amplitude patterns can form; here, these vary between wavenumber m = 5 at the top of the figure to m = 4 at the bottom. For k = 5, staggered oscillations with wavenumber m varying between 2 and 3. k = 6 stimuli can form non-alternating patterns with m = 6 (left) or alternating patterns with m = 3 (right). k = 10 stimuli tend to form alternating oscillations with m = 5.

though these and other spatiotemporal patterns shown in 2-D images from the network on a ring look very similar to snapshots of the network on the torus, these images of course include time. Small amplitude (e.g., 0.01 or 0.1) random initial conditions tended to initially look like the small-amplitude checkers, eventually evolving to the small-amplitude waves. We also find larger-amplitude oscillations: When given large (e.g., amplitude 1) cosine initial conditions, the network evolved to the alternating oscillations shown in (iii ). If we instead provide random initial conditions distributed on a larger interval (e.g., [-1, 1]), we can get (iv ), a staggered, wave-like large-amplitude alternating oscillation that is more difficult to classify. It appears like a slanted, wave-like alternating oscillation of m = 4 or 5; however, counting across a particular slice of time as we have for the other patterns results in 2-3 regions of high activity. We believe this is characteristic of the critical wavenumber 5, for reasons we explore below when we look at patterns induced when we turn on the spatial stimulus. Hereafter, we refer to this pattern as the (m =)2/3 pattern. Thus, already we see that the network exhibits tendencies to engage in wavelike, non-wavelike, and combinations of wavelike and non-wavelike behaviors. The other patterns we find just above a∗_ee are large- amplitude alternating oscillations at m = 3 and 4.

We now study the patterns that form in the network for aee= 5.8 < a∗ee ≈ 7.4 by applying

a cosine stimulus with wavenumber k and amplitude q (see Sec 3.2.2). For most permissible k (see Sec 3.2.2), the network displayed spatiotemporal patterns for q∗ ≤ q ≤ q∗ (we use

the same notation as in Sec 3.2.1.1, where it is explained in more detail). By following the steady state in q with AUTO, we see that Hopf bifurcations give rise to the patterns, all with nonzero modes m, as we see in Fig 17C. Thus, on the ring, q∗(k) and q∗(k) correspond to

Hopf bifurcations, lending support for our analogous classification of q∗ and q∗ on the torus

as Hopf bifurcations.

The network on the ring shows very similar spatiotemporal dynamics in the three different stimulus amplitude regimes that the network on the torus does. For q < q∗, the low steady

state is stable, and we observe low contrast stripes with wavenumber k, corresponding to the peaks and troughs of the stimulus. Similarly, for q > q∗, the high steady state is stable, leading to high contrast stripes of wavenumber k. In-between the TH bifurcations (q∗ ≤ q ≤ q∗), these steady states lose stability to oscillatory modes that depend on k.

We sampled the spatiotemporal modes by beginning the network with random initial conditions in a small interval of [wss

j − 0.01, wssj + 0.01] for w ∈ {u, v}, j ∈ {0, ..., N }, where

wss

j is the steady state value computed for a particular k, q combination (and is unstable

in between the Hopf bifurcations). For example, the bifurcation diagrams in Fig 17B show uss

0 for k ∈ {0, 5, 10, 12} and q ∈ [0, 1]. The network then evolved to steady spatially

heterogeneous oscillatory modes, some of which are shown in Fig 17C. We find several aspects of interest within this set of patterns. We note that these patterns are meant to be representative but not exhaustive.

For a constant stimulus (k = 0, Fig 17C), we find the network can evolve into two disparate types of patterns, one highly structured and one more amorphous and wave-like. The low-amplitude, highly structured pattern appears nearly identical to (i ). The large- amplitude, amorphous pattern shows the proclivity of the network to engage in wave-like behaviors as in (ii ) and (iv ). Note, too, the break up in the wave and the change in wavenumber m from 5 initially (i.e., at the top of the figure) to 4 (at the bottom of the figure, both of which are quite reminiscent of (iv ). Of course both patterns are also very similar to those we observed for k = 0 on the torus. We note also that driving the network with a stimulus of wavenumber k = 0 results in the most dramatic examples in which k and m differ. Otherwise, a k-stimulus tends to drive the network into patterns of either m = k (Fig 17C, k = 6, left) or m = k/2 (Fig 17C, k = 6, right), depending both on k and q. For example, all of the oscillatory patterns that we observed for stimuli with k = 10 and different q were alternating oscillations of mode m = 5. In contrast, k = 6 stimuli could drive the network into either alternating or non-alternating oscillations. k = 5 stimuli drive the network into more complicated patterns; in particular, they look like the alternating, staggered pattern m = 2/3 that we encountered above in (iv ) (Fig17A). Thus, we hypothesize that this pattern (here and in iv ) arises due to the tendency of the network to engage in alternating oscillations and in wavelike patterns; since 5 is odd, the network settles on a wavelike, staggered oscillatory pattern with wavenumbers of 2 or 3.

3.2.2.2 Resonance on the ring In Fig 17B, we can clearly see a similar resonant