Decomposition of spatial information

We shall continue our analysis by further decompose the structural information into contributions from position and length scales. For simplicity, let us assume that we have a chemical system, characterised by concentrations ci(x, t) for the different molecules that are normalised at each position x,

 

         clip_image003 .                                                                                          (9)

 

In order to be able to analyse contributions from different length scales, we introduce a resolution dependent concentration clip_image006, by convolution of the original one ci with a Gaussian,

 

         clip_image009                                                           (10)

 

We will use the last expression as the resolution operator, but when calculating clip_image011, the first expression will be used. We assume that this operation handles the boundary conditions so that in the limit of r ® ¥, i.e., complete loss of position information, the concentrations equal the average concentrations in the system,

 

         clip_image014.                                                                                          (11)

 

The derivation of a decomposition of the total information K into different contributions will now be slightly different from the one in Chapter 6, since the distributions are now normalised in each position (instead of over the system volume). Our starting point for decomposing the total information K,

 

         clip_image017                               (12)

 

illustrates that the chemical information Kchem can be detected regardless of how bad the resolution is. This is obvious, since complete loss of resolution results in average concentrations in the system which determines the chemical information. This means that the spatial information Kchem can be decomposed as follows

 

     clip_image020        (13)

 

In the last step a partial integration is invoved, including an assumption of periodic (or non-flow) boundary conditions. The equilibrium concentration ci0 in the last expressions could be removed (as it disappears under the differential operation), but we keep it as it will serve a purpose in later derivations. In the last expression we recognize a local information density, k(r, x), which can be written in three different forms, of which the last one will be used later,

 

         clip_image023                                                   (14)

 

 

The information density k(r, x) can be integrated over space so that we achieve the spatial information at a certain length scale,

 

         clip_image026.                                                                           (15)

 

A schematic illustration of this decmposition is shown in the figure below, illustrating a chemical patterns in a two-dimensional space with its variations along the axis of resolution r.  

 

 

         clip_image029

 

Figure 1.  An illustration of a two-dimensional chemical pattern at different levels of resolution r, along with a schematic illustration on how the spatial information may be decomposed between different length scales. The chemical information can be detected regardsless of how bad the resolution is and it is therefore located at infinite r.