Let $(R,\mathfrak{m})$ be a (not necessarily commutative) local ring with commutative residue field $k$. Define it's betti numbers as $$\beta_i(R):=\dim_k\mathrm{Tor}^i_R(k,k).$$
The associated graded ring
$$\mathrm{gr}(R)=\bigoplus_{n\geq 0}\mathfrak{m}^n/\mathfrak{m}^{n+1}, $$ is also a local ring with residue field $k$. One can easily see that for $i=0,1$, we have $$\beta_i(R)=\beta_i(\mathrm{gr}(R)).$$
Is there any connection between the betti numbers $\beta_i(R)$ and $\beta_i(\mathrm{gr}(R))$ for $i\geq 2$?