We now apply this network model to a network of cells found in a hypercolumn in primary visual cortex. We divide the network neurons into two types: excitatory and inhibitory. Each cell is selective to the orientation of the visual stimulus and is parameterized by its preferred orientation (PO). For excitatory neurons, the PO for cell ii is given by θi=π2+iπNexθi=π2+iπNex, and the reversal potential is uniform and denoted EexEex. The PO and reversal potential for inhibitory neurons are defined analogously.
The peak conductance at a network synapse is dependent on the PO of the pre and postsynaptic neurons. If cell jj is excitatory, then the peak conductance at the excitatory synapse from cell jj to cell ii is given by
G
i
j
(

θ
i

θ
j

)
=
G
¯
e
x
λ
e
x
exp
(

θ
i

θ
j

/
λ
e
x
)
,
G
i
j
(

θ
i

θ
j

)
=
G
¯
e
x
λ
e
x
exp
(

θ
i

θ
j

/
λ
e
x
)
,
(10)where λexλex is the excitatory decay constant in space. The peak conductance at inhibitory synapses is defined analogously. We assume that no cell synapses onto itself, or Gii=0∀iGii=0∀i, and that no inhibitory to inhibitory synapses exist. Figure 2 shows the network architecture for a small network.
Excitatory neurons in the lateral geniculate nucleus (LGN) are the external sources for our network in the primary visual cortex. The total mean firing rate of all external sources to cell ii is given by
f
L
G
N
(

θ
i

θ
0

)
=
f
¯
L
G
N
C
[
(
1

ϵ
)
+
ϵ
cos
(
2
(
θ
i

θ
0
)
)
]
,
f
L
G
N
(

θ
i

θ
0

)
=
f
¯
L
G
N
C
[
(
1

ϵ
)
+
ϵ
cos
(
2
(
θ
i

θ
0
)
)
]
,
(11)where θiθi is the PO of cell ii, and θ0θ0 is the orientation of the stimulus. The parameter CC dictates the stimulus contrast, and the parameter ϵϵ measures the degree of tuning of the LGN input. The maximum firing rate, f¯LGNf¯LGN, is reached if C=0C=0 and θi=θ0θi=θ0. Note that if ϵ=0ϵ=0, the LGN input is untuned, and if ϵ=0.5ϵ=0.5, the LGN input vanishes when θiθ0=π2θiθ0=π2.
To ensure that we have understood and implemented the model as Shriki intended, we now test our model by reproducing the conductancebased portion of Figure 3 in Shriki's paper [1]. For this simple example, the network in the primary visual cortex consists of NN excitatory, homogeneous neurons. The homogeneity of the network means that each neuron is connected to every other neuron with peak conductance GG. Also, each neuron has the same PO, implying that the mean firing rate, fLGNfLGN, is uniform across all neurons. To model this, we generate NN uncorrelated Poisson spike trains with mean firing rate fLGNfLGN. We now implement the model for this homogeneous network to show how the firing rate depends on the peak synaptic conductance, G.G. Our results are shown in Figure 3.
Table 2: Parameter values used for figure 3
N
e
x
N
e
x

1000 
G
e
x
t
G
e
x
t

0.0035 mS/cm2mS/cm2 
f
L
G
N
f
L
G
N

1570 Hz 
λ
e
x
λ
e
x

5 rad 
τ
e
x
τ
e
x

5 ms 
τ
e
x
t
τ
e
x
t

5 ms 
V
r
e
s
t
V
r
e
s
t

73 mV 
C
m
C
m

1 μF/cm2μF/cm2 
E
L
E
L

65 mV 