Quasicrystal patterns in a neural field model

Doubly periodic patterns in planar neural field models have been extensively studied since the 1970s for their role in explaining geometric visual hallucinations. The study of activity patterns that lack translation invariance has received little, if any, attention. Here we show that a scalar neural field model with a translationally invariant kernel can support quasicrystal solutions and that these can be understood using many of the theoretical tools developed previously for materials science. Our approach is constructive in that we consider constraints on the nonlocal kernel describing interactions in the neural field that lead to the simultaneous excitation of two periodic spatial patterns with incommensurate wavelengths. The resulting kernel has a shape that is a modulation of a Mexican-hat kernel. In the neighborhood of the degenerate bifurcation of a homogeneous steady state, we use a Fourier amplitude approach to determine the value of a Lyapunov functional for various periodic and quasicrystal states. For some values of the parameters defining a translationally invariant synaptic kernel of the model, we find that quasicrystal states have the lowest value of the Lyapunov functional. We observe patterns of 12-fold, 10-fold, and 6-fold rotational symmetry that are stable, but none with 8-fold symmetry. We describe some of the visual hallucination patterns that would be perceived from these quasicrystal cortical patterns, making use of the well known inverse retinocortical map from visual neuroscience.

Figure 1

Series of plots showing simulations of the integral model (1) at (a) $t=0$, (b) $t=40$, (c) $t=100$, (d) $t=120$, (e) $t=140$, and (f) $t=300$, posed on a large Euclidean plane with a sigmoidal firing rate function given by Eq. (3) and connectivity function given by Eq. (9). The simulations are initiated with a localized spot perturbed with a 10-fold symmetric function (see Appendix pp4 for further details). The parameters are $\beta =0.39$, $h=2.371$, $b_1=0.691$, $b_2=0.619106$, ${\alpha }_1=2.144141$, ${\alpha }_2=0.518136$, $s_1=0.574835$, and $s_2=0.236861$.

Figure 2

Plot of the Lyapunov functional $E\left[u\right]$, given by Eq. (4), during the evolution of a decagonal quasicrystal in time. Snapshots of the evolution are recreated for easier referencing. Each layer of growth around a central localized quasipattern is accompanied by a significant decrease in $E\left[u\right]$. The parameters are the same as in Fig. 1.

Figure 3

Connectivity function and its Fourier transform for the neural field model (1). (a) Radially symmetric connectivity kernel given by Eq. (9) as a function of distance $r$, with the parameters $\beta =9$, $h=0.06$, $b_1=0.681$, $b_2=0.655$, ${\alpha }_1=0.08036$, ${\alpha }_2=0.016238$, $s_1=0.572164$, and $s_2=0.211759$. (b) Fourier transform as a function of wave number $k$, showing global maxima at the two critical wave numbers $k=1$ and $k=q$.

Figure 4

Space-time profiles of the field $u(\mathbf{r},t)$ at a fixed large time showing (a) dodecagonal, (b) decagonal, and (c) octagonal quasicrystal structures for the planar neural field model (1) (on a large domain $[-L,L]\times [-L,L]$ where $L=36\pi$) and (d)–(f) the corresponding Fourier transforms. (d)–(f) show two critical circles in the power spectra for (a)–(c). The parameters are (a) $q=2cos(\pi /12)$, $\beta =7.9$, $h=0.07$, $b_1=1.1$, $b_2=0.69$, ${\alpha }_1=0.05$, ${\alpha }_2=0.183$, $s_1=0.4931$, and $s_2=0.7711$; (b) $q=2cos(\pi /5)$, $\beta =9$, $h=0.06$, $b_1=0.681$, $b_2=0.655$, ${\alpha }_1=0.08036$, ${\alpha }_2=0.016238$, $s_1=0.572164$, and $s_2=0.211759$; and (c) $q=2cos(\pi /8)$, $\beta =4.1$, $h=0.09$, $b_1=0.76$, $b_2=0.675$, ${\alpha }_1=0.1692$, ${\alpha }_2=0.1215$, $s_1=0.5812$, and $s_2=0.4172$.

Figure 5

Phase diagram of the lowest value of the Lyapunov functional steady-state solution of the neural field model (1) for $q=2cos(\pi /5)$. Here DQC and HEX represent decagonal and hexagonal structures, respectively; the DDQC, DQC, OQC and HEX patterns have the same (zero) energy in the black region. The parameters are $b_1=0.681$, $b_2=0.655$, ${\alpha }_1=0.08036$, ${\alpha }_2=0.016238$, $s_1=0.572164$, $s_2=0.211759$, and $\epsilon =0.1$.

Figure 6

Perception of (a) dodecagonal, (b) decagonal, and (c) octagonal quasicrystals patterns of neural activity in V1, as presented in Fig. 4, constructed via the inverse retinocortical map.