The typical form sets the curvature at both ends to zero (the natural boundary condition). The coefficients come from a system of equations over the values at all the points, so when one value changes, the curve in distant intervals moves as well.
For curve interpolation, the form covered here is applied to the log discount factor (interpolating the discount factor is the traditional approach, but there are also methods that interpolate the forward rate). The forward rate can turn negative. Le Floc'h's paper says that, of the Hagan and West criteria, a standard cubic spline fails the criteria that the forward rate be positive and that the hedge allocation stay with nearby maturities. A cubic Hermite interpolation, which sets the slope at each point from the values of the neighboring intervals (a monotonicity-preserving interpolation, for example), narrows the range over which a change spreads.