 Solution

The common vector field $F$ represents an identical reference cell. The small cell-specific term $ϵG_i$ represents heterogeneity of cellular properties, such as differing ion-current parameters or intrinsic oscillation rates. It is not external random noise: the displayed model is deterministic.

The Kronecker delta factors restrict coupling to the first state component, the membrane potential. Substitution gives a coupling current in the voltage equation proportional to $ϵ{\sum }_j{d}_{ij}(u_j^1-u_i^1)$, while the other state components have no direct intercellular coupling. This is the linear voltage-difference response of gap junctions. Gating and chemical variables can still change indirectly through their own coupled intracellular dynamics. The zero diagonal removes self-coupling, which would contribute zero anyway. Physical reciprocal gap-junction coupling is often symmetric, but zero diagonal alone does not impose reciprocity; that distinction matters in the final part.