Modeling spatial patterns in the visual cortex

We propose a model for the formation of patterns in the visual cortex. The dynamical units of the model are Kuramoto phase oscillators that interact through a complex network structure embedded in two dimensions. In this way the strength of the interactions takes into account the geographical distance between units. We show that for different parameters, clustered or striped patterns emerge. Using the structure factor as an order parameter we are able to quantitatively characterize these patterns and present a phase diagram. Finally, we show that the model is able to reproduce patterns with cardinal preference, as observed in ferrets.

Figure 1

A lattice with $N=60\times 60$ sites, showing explicitly the connections of nine units. The background presents a typical pattern representing the phase value of the units.

Figure 2

Mexican hat interaction function. Parameter $C$ modulates the balance between excitatory and inhibitory interactions, while $\sigma$ regulates the distance at which there is an effective interaction.

Figure 3

Temporal evolution of the phases for four different sets of parameters. The snapshots correspond to times $t=0,25,50$, and 100 s. The parameters for (a) and (b) are $A=10$, $\alpha =2.1$, $m=100$, $M=500$, $C=1$; additionally in (a) ${\sigma }^2=6$ and in (b) ${\sigma }^2=14$. For (c) $A=10$, $\alpha =2.5$, $m=4$, $M=400$, $C=1$, ${\sigma }^2=6$, and in (d) $A=10$, $\alpha =5$, $m=100$, $M=500$, $C=2$ and ${\sigma }^2$ is variable.

Figure 4

Snapshots of four stationary states of the system (left column) and their corresponding structure factors (right column). System size is $N=140\times 140$ and parameters are the same as in Fig. 3.

Figure 5

Mean value of the structure factor as a function of $k$ vector, averaged over all directions. Different qualitative behaviors corresponding to the striped phase (top curve) and clustered phase (bottom curve) can be clearly distinguished. System size is $N=140\times 140$ and parameters are the same as in Fig. 3 and 3.

Figure 6

Structure factor as a function of $k_x$ and $k_y$ for parameters that presents a clear orientational preference characterized by the presence of two peaks that can be clearly distinguished.

Figure 7

Mean value of the structure factor as a function of $k$ for two directions ($k_x,k_y$). The curves reflect the difference between the strength of the structure factor in some directions for the case presented in Fig. 6.

Figure 8

Phase diagram varying the Mexican hat parameters $C$ and $\sigma$. The transition between striped and clustered phases was determined quantitatively using the structure factor.

Figure 9

Histograms of the phases for ${\sigma }^2=6$ constant and varying C (from top to bottom $C=0.5$, $1.0$, $2.0$, and $3.0$). The parameters used are $A=10$, $\alpha =2.1$, $m=100$, and $M=500$. System size is $N=140\times 140$.