Hi!

I have a first ODESystem with a parameter `gNa`

and a second ODESystem with a variable `g(t)`

. How to create a third system that links the two together so that every time `gNa`

appears in the first system we actually take `g(t)`

?

Note: I want to preserve the first system which has `gNa`

as a parameter, don’t want to define it as a variable from the start. Just want to extend it when it has dynamics.

Here’s a minimally working example that reproduces the following error.

```
using ModelingToolkit
const t = Num(Sym{Real}(:t))
const D = Differential(t)
struct Na{T}
gNa::T
RNa::T
mNa::T
hNa::T
end
m∞(::Na, V) = 1.0 / (1.0 + exp((V + 25.5) / -5.29))
h∞(::Na, V) = 1.0 / (1.0 + exp((V + 48.9) / 5.18))
function channel_dynamics(ch::Na)
states = @variables mNa(t) hNa(t) INa(t) V(t)
parameters = @parameters gNa ENa
eqs = [D(mNa) ~ (m∞(ch, V) - mNa),
D(hNa) ~ (h∞(ch, V) - hNa),
INa ~ gNa * mNa^3 * hNa * (ENa - V),
D(V) ~ INa]
defaultmap = [mNa => ch.mNa, hNa => ch.hNa, V => -50., gNa => ch.gNa]
return eqs, states, parameters, defaultmap
end
function regul_indep(ch::Na)
states = @variables g(t) R(t) Ca(t)
parameters = @parameters τch Ca_tgt τg
eqs = [D(R) ~ (Ca_tgt - Ca) / τch,
D(g) ~ (R - g) / τg,
D(Ca) ~ (-Ca + 0.5)]
defaultmap = [g => ch.gNa, R => ch.RNa, Ca =>2.0]
return eqs, states, parameters, defaultmap
end
ch = Na(70.0, 70.0, 0.0, 0.0)
eqs, states, parameters, defaultmap = channel_dynamics(ch)
@named sys1 = ODESystem(eqs, t, states, parameters; defaults = defaultmap)
eqs, states, parameters, defaultmap = regul_indep(ch)
@named sys2 = ODESystem(eqs, t, states, parameters; defaults = defaultmap)
sys = extend(sys1, sys2)
append!(equations(sys), [sys1.ps[1] ~ sys2.states[1]]) #gNa ~ g(t)
final = alias_elimination(sys)
final = structural_simplify(final)
```

When I add the extra equation `[gNa ~ g(t)]`

to `sys`

I can see the equations are all OK with alias_elimination. The problem is with structural_simplify, which gives the error

`ERROR: LoadError: ExtraEquationsSystemException: The system is unbalanced. There are 7 highest order derivative variables and 8 equations.`

I found this related issue but doesn’t seem to be solved: Redundant equations causes a crash · Issue #936 · SciML/ModelingToolkit.jl · GitHub

Over-specifying the system with more equations shouldn’t be a problem. What’s the correct way to do this?