4.1.2 Connectivity and coupling weight

In local circuits, the connectivity between two groups of neurons (LGN, 4E, 4I, 6E, 6I) is given by a truncated 2D Gaussian kernel. The truncated Gaussian is then multiplied by the synaptic coupling weight that is independent of location. L6 E-neurons have long range connections, which will be specified later. Take 4E-to-4E connections as an example. The coupling between two groups of neurons with coordinates z and z* in Ω is given by

Here W_4E4E is the coupling weight, σ_4E4E is the standard deviation of the Gaussian kernel, and R_4E4E is the radius of the footprint of the 4E-to-4E connection. Other coupling weights are given analogously.

As shown in Fig 2A, 4E and 4I receive connections from LGN, 4E, 4I, and 6E; 6E and 6I receives connections from LGN, 4E, 6E, and 6I. See Table A-E of S1 Text for the full list of parameters. The L6-to-L4 coupling weight W_6E4E is a control parameter. We choose different values of W_6E4E from 0.44 to 0.48 in our simulations.

In addition to local connections, 6E neurons also have long-range connections to L6 neuron groups with the same orientation preferences. We assume local circuits have radius 0.15. Setting p_E(0.15) = p_I(0.15) = 1.0, the coupling weight between two L6 E-groups with the same orientation preferences and located at z and z* with |z − z* | > 0.15 is

where q₀ is chosen as the connection probability for long-range and local circuits match up at |z − z* | = 0.15, π(z,z*) = 1 if and only if z and z* has the same orientation preference (and equals 0 otherwise). The long-range 6E-to-6I connection function, p_I(d), is defined analogously. We have assumed also that p_E(1.6) = p_I(1.6) = 0. See Table E of S1 Text for the table of values of p_E(d) and p_I(d) at different interpolation points. Functions p_E(d) and p_I(d) are piecewise linear between those interpolation points.

It remains to give the coupling weight from the ambient drive, denoted by Amb_4E,Amb_4I, Amb_6E, and Amb_6I. We assume the ambient drive to each type of neuron group is a constant that is small comparing with other terms. Parameters of ambient drives can be found in Table A-D of S1 Text.