Properties of associated Boolean functions of quadratic APN functions

For a function $F:\mathbb{F}_2^n\to \mathbb{F}_2^n$, it is defined the associated Boolean function $\gamma_F$ in $2n$ variables as follows: $\gamma_F(a,b)=1$ if $a\neq\mathbf{0}$ and equation $F(x)+F(x+a)=b$ has solutions. A vectorial Boolean function $F$ from $\mathbb{F}_2^n$ to $\mathbb{F}_2^n$ is called almost perfect nonlinear (APN) if equation $F(x) + F(x + a)=b$ has at most $2$ solutions for all vectors $a,b\in\mathbb{F}_2^n$, where $a$ is nonzero. In case when $F$ is a quadratic APN function its associated function has the form $\gamma_F(a,b) = \Phi_F(a) \cdot b + \varphi_F(a) + 1$ for appropriate functions $\Phi_F:\mathbb{F}_2^n\to \mathbb{F}_2^n$ and $\varphi_F:\mathbb{F}_2^n\to \mathbb{F}_2$. We study properties of functions $\Phi_F$ and $\varphi_F$, in particular their degrees.