Using Gauss's law, the evoked potential measured at the epidural screw electrodes can be represented as a function of the summation of current dipoles generated at the visual cortex multiplied by a constant that represents the surface area and the conductivity of the medium
34:
\begin{eqnarray}V = \sum {{k}_{cortex}}I\end{eqnarray}
where
kcortex is a constant that is inversely proportional to the conductivity of the medium and the distance from the current source
I to the recording location. Meeren et al.
12 demonstrated that cortical N1 is generated by excitatory postsynaptic potentials at layers 5 and 6 of the neocortex. This suggests that the mean transmembrane potential that generates N1 can be modeled as directly proportional to the number of axons in the optic radiation carrying action potentials from the dLGN (
NdLGN).
35 \begin{eqnarray}\bar{I} = qP{{N}_{dLGN}}\end{eqnarray}
where
q is a quantal amplitude generated by the release of neurotransmitters by the presynaptic neuron, and
P is the probability that each axon will release neurotransmitters when it receives an action potential. Ultimately,
kcortex,
q, and
P do not depend on stimulation of the eye, which means that the N1 amplitude measured at the epidural screw electrode is approximately proportional to the number of dLGN neurons that have generated an action potential.
\begin{eqnarray}V = k{{N}_{dLGN}}\end{eqnarray}
where
k is a constant that depends on
kcortex,
q, and
P and remains roughly invariant within a single experimental session lasting less than 20 minutes. The relationship between the number of retinal ganglion cells (RGCs) carrying the action potential to the number of dLGN neurons generating an action potential is far from simple. The exact physics of synaptic connectivity is complex, with asymmetry in the ipsilateral and contralateral projections to the dLGN. We make two careful assumptions to model this scenario. First, the number of dLGN neurons generating action potentials is monotonically dependent on the number of RGCs, meaning that a greater number of RGCs will generate the same or a greater number of post-synaptic action potentials.
\begin{eqnarray}{{N}_{dLGN}} = f( {{{N}_{RGC}}} )\end{eqnarray}
where
NRGC is number of axons in the optic nerve carrying the action potential from the RGCs to the dLGN, and
f(
x) is a monotonic function that describes the relationship between the number of RGCs and dLGN that carry action potentials. Second, we introduce a variable (
x), defined as the proportion of RGCs that synapses to contralateral dLGNs compared to total RGCs that synapse to either ipsilateral or contralateral dLGNs. Using this, we generated four equations:
\begin{eqnarray}{{V}_{R{{F}_{LS}}}} = {{k}_{contralateral}}*x*f( {{{N}_{RG{{C}_{OD}}}}} )\end{eqnarray}
\begin{eqnarray}{{V}_{R{{F}_{RS}}}} = {{k}_{ipsilateral}}* ( {1 - x} )*f ( {{{N}_{RG{{C}_{OD}}}}} )\end{eqnarray}
\begin{eqnarray}{{V}_{L{{F}_{LS}}}} = {{k}_{ipsilateral}}* ( {1 - x} )*f ( {{{N}_{RG{{C}_{OS}}}}} )\end{eqnarray}
\begin{eqnarray}{{V}_{L{{F}_{RS}}}} = {{k}_{contralateral}}*x*f ( {{{N}_{RG{{C}_{OS}}}}} )\end{eqnarray}
where
k is the constant of proportionality of evoked potential to
NdLGN, and
\({{N}_{RG{{C}_{OD}}}}\) and
\({{N}_{RG{{C}_{OS}}}}\) represent the number of RGCs carrying the action potential in the right and left optic nerves, respectively. Also,
k was divided into
kcontralateral and
kipsilateral because of the aforementioned topographical differences between cortical responses to contralateral and ipsilateral eye stimulations which result in different synapse kinetics and different conductance values between the ipsilateral and contralateral zones of responses to the screw electrode. Rearranging the above equations, we obtain:
\begin{eqnarray}
\frac{{f( {{{N}_{RG{{C}_{OD}}}}} )}}{{f( {{{N}_{RG{{C}_{OS}}}}} )}} = \sqrt {\frac{{{{V}_{R{{F}_{LS}}}}*{{V}_{R{{F}_{RS}}}}}}{{{{V}_{L{{F}_{LS}}}}*{{V}_{L{{F}_{RS}}}}}}} \end{eqnarray}