Validation of soma network with a gap juncion
A network of 2 identical somas with nax, kamt and kdrmt mechanisms in 3 configurations:
- Unconnected
- Connected by a single gap junction
- Connected by a single gap junction and with tweaked nax parameter on one of the somas
Built in Arbor, and Neuron and solved numerically using Julia.
The set of differential equations governing the current and voltage of the somas are:
dV/dt = I(m0, h0, m1, h1, m2, V) (1)
dm0/dt = (m0inf(V)-m0)/m0tau(V) (2)
dh0/dt = (h0inf(V)-h0)/h0tau(V) (3)
dm1/dt = (m1inf(V)-m1)/m1tau(V) (4)
dh1/dt = (h1inf(V)-h1)/h1tau(V) (5)
dm2/dt = (m2inf(V)-m2)/m2tau(V) (6)
where m0, h0 control the current changes of nax; m1, h1 control kamt; m2 controls kdrmt.
The Neuron and Julia solutions use the cvode solver from the Sundials package to solve the ODE system.
The Arbor solution decouples equation (1) from the rest, making the assumption that the m and h variables are not a function of voltage, to solve equations (2)-(6) exactly. This is a reasonable assumption for small enough time steps.
Gap Junctions are modeled as a conductance between 2 compartments (somas).
Arbor handles gap junctions using one of 2 methods:
- Using an implicit method: the voltage of the 2 connected somas is calculated simultaneously in the same procedure at every time step. This means that the contribution of the gap junctions is part of equation (1).
- Using an explicit method: The current and voltage of the 2 somas are solved separately at every time step; the current of the gap-junction is calculated from the potential difference of the two somas and that current is used as a contribution to the soma currents for the next iteration.
Neuron handles gap junctions using one of 2 methods:
- Using an explicit method: Similar to Arbor's explicit method.
- Adding the linear equation of the gap-junction to the tree matrix current balance equations; i.e. adding a new differential equation to the system of ODEs solved by cvode.
We have tested all 4 methods in Arbor and Neuron.
For Arbor, explicit and implicit methods give very similar results. The results displayed in this document are the results of the implicit method, but the explicit method could have been used with minor changes.
For Neuron, the results of using the cvode method match the Julia solution more closely than the explicit method. We use the cvode results in this document.
We present the graphs of the 3 models in Arbor, Neuron and Julia.
Two identical somas, with nax, kamt, and kdrmt mechanisms; subjected to a continuous current stimulus that is delayed for 10ms for one of the two somas.
Zoomed in:
Two identical somas, with nax, kamt, and kdrmt mechanisms; subjected to a continuous current stimulus that is delayed for 10ms for one of the two somas.
Zoomed in:
Two somas, with nax, kamt, and kdrmt mechanisms - with different nax properties; subjected to a continuous current stimulus that is delayed for 10ms for one of the two somas.
Zoomed in:
