yeah, i don't find it perfectly clear what they do. I based my impression on this:

All light stimuli LVk are updated synchronously on the basis of the following recurrent neural network dynamics [14,17,18], which was derived from the Hopfield–Tank–Amari model [21].

LVk(t+Δt)=1−σ1000,−0.5(∑ WVk,Ul⋅σ35,0.6(XUl(t))),

σγ,θ=1/(1+exp(−γ⋅(x−θ))),

WVk,Ul=

−λ(if V=U and k≠l)

−μ(if V≠U and k=l)

−ν⋅dist(V,U)(if V≠U and |k−l|=1),

0(otherwise).

The sigmoid function σ35,0.6 adjusts the sensitivity of the control, σ1000,−0.5 performs similar to the step function and the inhibitory coupling weight WVk,Ul (= WUl,Vk < 0) results in the decrease in XUl with the increase in XVk and vice versa. Parameters λ = 0.5, μ = 0.5 and ν impose the following constraints on TSP, respectively: (i) prohibition of revisiting a once-visited city, (ii) prohibition of simultaneous visits to multiple cities and (iii) minimization of the total distance, where dist(V, U) is the distance between cities V and U. To maximize the effect of (iii), it is necessary to set ν as large as possible so that differences in inter-city distances can be maximally amplified. However, ν must be below an upper limit ν⋆, otherwise some branches are illuminated even when they select some possible tours. The value of the limit ν⋆=−θ/(dist(V⋆,V′⋆)+dist(V′⋆,V′′⋆)) is obtained numerically, where θ = −0.5 and {V⋆,V′⋆,V′′⋆}=argmax{V,V′,V′′}(dist(V,V′)+dist(V′,V′′)). Thus, λ, μ and ν can be automatically determined for given a map. For the eight-city map, we set ν to be ν⋆≃0.008197>ν=0.0081. Similarly, for the four-, five-, six- and seven-city maps, we set ν = 0.00495, 0.00565, 0.0067 and 0.0076, respectively.

It reads to me as a function that encodes the problem.

W is matrix that represents the problem instance - which channels may be active at the same time, which channels are consecutive to each other and what is their distance. X is the current state vector.

The numbering is: Vk is a channel representing a visit to city V in the k-th slot of the trip. Same for Ul, mutatis mutandis.

XUl is the fraction of channel Ul filled by the amoeba.