Radial density calculations

In order to assess the local neuronal density in the vicinity of $A\beta$ deposits, we borrow from statistical physics the concept of a radial density function, $\rho(R_i,r)$, that in our present study is defined as the number of neurons per unit area inside a ring of radius $r$ from an origin $R_i$, which is the center of an $A\beta$ deposit. A similar concept of local density calculations was introduced in ref. [31] for the description of neuronal microcolumns in the human cortex. For example, using the $A\beta$ deposit within the neuronal field shown in Fig. 1 (b), the origin for $\rho(R_i,r)$ is at the center of an A$\beta$ deposit and a ring is defined with inner and outer radius corresponding to $r$ and $\delta r + r$, respectively. $\rho(R_i,r)$ at distance $r$ from the origin is defined as the number of neurons within the ring divided by the area of the ring. In practice, a computer algorithm calculates $\rho(R_i,r)$ using as input the two-dimensional $x$-$y$ maps of neuronal coordinates obtained using the methods from section IIB. We then calculate the average of the radial density function over all $A\beta$ deposits according to $\rho(r)=(1/N)\sum^N_{i=1}\rho(R_i,r)$, where $N$ is the number of A$\beta$ deposits.

The strength of analyzing $\rho(r)$ is that it can test predictions of models of $A\beta$ effects on neurons by quantitating disruptions in the neuronal distribution in the vicinity of each $A\beta$ deposit. In particular, it allows us to test four models of the effect of $A\beta$ deposition (Fig. 2):

1.

Non-toxic diffuse lesion: The neuronal density will be uniform both within the $A\beta$ deposit and in the immediate vicinity, resulting in a horizontal line as the $\rho(r)$ function (Fig. 2(a)).

2.

Non-toxic mass lesion: The $A\beta$ deposit displaces neurons, resulting in a reduced neuronal density within the $A\beta$ deposit, a local increase in $\rho(r)$ representing neurons pushed to the periphery, followed by a uniform $\rho(r)$ thereafter (Fig. 2(b)).

3.

Focally toxic mass lesion: The $A\beta$ deposit kills neurons immediately within the $A\beta$ deposit, resulting in a reduced neuronal density within the $A\beta$ deposit, followed by a uniform $\rho(r)$ afterwards (Fig. 2(c)).

4.

Toxic mass lesion with toxic penumbra: The $A\beta$ deposit kills neurons resulting in a reduced radial density both within and in the vicinity of the $A\beta$ deposit (Fig. 2(d)).