Spatiotemporal canards in neural field equations

Canards are special solutions to ordinary differential equations that follow invariant repelling slow manifolds for long time intervals. In realistic biophysical single-cell models, canards are responsible for several complex neural rhythms observed experimentally, but their existence and role in spatially extended systems is largely unexplored. We identify and describe a type of coherent structure in which a spatial pattern displays temporal canard behavior. Using interfacial dynamics and geometric singular perturbation theory, we classify spatiotemporal canards and give conditions for the existence of folded-saddle and folded-node canards. We find that spatiotemporal canards are robust to changes in the synaptic connectivity and firing rate. The theory correctly predicts the existence of spatiotemporal canards with octahedral symmetry in a neural field model posed on the unit sphere.

Figure 1

Time simulations of the system (4) for $\varepsilon =3.62\times {10}^{-3}$, $\beta =\gamma =0$ and (a) $\alpha =0.49$, (b) $\alpha =0.50$, and (c) $\alpha =0.51$; $W$ is as in Eq. (9) with $a=\mathrm{\Lambda }=1$ and $b=0.3$; $\mathrm{\Theta }\left(u\right)\approx 1/[1+exp(-50u\left)\right]$. In (b) we superimpose the threshold crossings $x=\pm \xi \left(t\right)$ on the pattern, shown for $t\in [12.5,25]$ on a lighter background for better contrast. (d) Examples of the functions ${\psi }_i$ corresponding to kernels $W_i$, $i=1,2,3$, commonly used in neural field models: $W_1(x,y)=(1+0.5|x-y\left|\right)exp(-|x-y\left|\right)$ is a purely excitatory, translation-invariant kernel used, for instance, in Ref. [25]; $W_2(x,y)=exp(-0.25|x-y\left|\right)(0.25sin|x-y|+cos|x-y\left|\right)$ is an excitatory-inhibitory, oscillatory, translation-invariant kernel used in Refs. [26, 27]; $W_3$ is the oscillatory heterogeneous kernel (9) used in (a)–(c) and in all other calculations of this paper [28, 29, 30]. We plot $\xi$ on the vertical axis, so that the figure can be read as a bifurcation diagram of the full neural field system (1). Solid (dashed) lines indicate stable (unstable) stationary patterns.

Figure 2

Examples of solutions containing spatiotemporal canards of folded-saddle type. (a) Time simulation of the folded-saddle case for $\alpha =0.5$, $\beta =\gamma =0$, and remaining parameters as in Fig. 1. (b) Solution to the full spatiotemporal model (4), also shown in (a) and in Fig. 1, projected on the $(h,q,\xi )$ phase space (blue), where we also plot the critical manifold $S^0$ (gray) and singular canards of (7) on $S^0$ (red). (c) Projection on the $(q,\xi )$ plane, showing the attracting ($A$) and repelling ($R$) sheets of $S^0$, revealing a spatiotemporal canard.

Figure 3

Additional features of the folded-saddle scenario. In (a) and (b) we repeat the simulation of Fig. 2 with $\varepsilon =3.6\times {10}^{-3}$, finding a periodic solution displaying a jump-on canard. (a) Projection of the trajectory of the full spatiotemporal model on the $(h,q,\xi )$ space (blue), where we also plot the critical manifold $S^0$ (gray) and singular canards on $S^0$ (red). (b) Projection on the $(q,\xi )$ plane with attracting ($A$) and repelling ($R$) sheets of $S^0$. In (c) and (d) we show a family of orbit segments passing near the folded saddle and displaying sensitivity to initial conditions, separating orbits jumping up toward ($A$) or jumping down toward another attracting sheet of $S^0$ [also marked as ($A$) in panel (b)].

Figure 4

Examples of solutions containing spatiotemporal canards of folded-node type. Parameters are as in Fig. 2, except $\alpha =\gamma =1$, $\beta =0$, for which the theory predicts folded-node canards. In (a)–(c) we plot the full spatiotemporal solution (a), its projection on the $(h,q,\xi )$ space (b), and on the $(q,\xi )$ plane (c); the latter show oscillations typical of folded-node canards, and therefore correspond to spatiotemporal folded-node canards in the neural field model. In (d)–(i) we set $\varepsilon =3.6\times {10}^{-3}$, $\alpha =1$, $\gamma =0.7$, and retain all other parameter values; when initial conditions are varied slightly, a variable number of small oscillations is found near the folded node, as expected from the ODE theory. We set $q\left(0\right)=-18.35$ [label 1 in (d), (e), (f), and (i)] and $q\left(0\right)=-18.40$ [label 2 in (f), (g), (h), and (i)]. (f) Projections on the $(q,\xi )$ plane, revealing an initial drift near the folded node, during which trajectory 1 (2) displays 3 (5) small-amplitude oscillations around the folded singularity [see inset (i)].

Figure 5

Spatiotemporal canards of folded-saddle type occurring in a neural field model posed on a spherical domain using $h$ as the continuation parameter. (b) Bifurcation diagram of steady states with octahedral symmetry in the system (11). (a,c) Bifurcation diagram (red), orbit (blue), and representative patterns obtained when $h$ varies slowly through (a) a low-lying fold and (c) one of the higher folds. When the evolution of $h$ is decoupled from $u$, we observe spatiotemporal canards of folded-saddle type. Animations displaying a canard cycle and orbits jumping to the upper and lower branch can be found in the Supplemental Material [39].