Notice that this is just the number of paths not passing through the middle point. We will use complementary counting. There are $\binom42 = 6$ paths from $A$ to the middle point, and $\binom42 = 6$ paths from the middle point to $B.$ So, there are $6^2 = 36$ paths from $A$ to $B$ that don't work. The total number of paths from $A$ to $B$ is $\binom84 = \frac{8 \cdot 7 \cdot 6 \cdot 5}{4 \cdot 3 \cdot 2 \cdot 1} = 70,$ so the answer is $70 - 36 = \boxed{34}.$