I received a link to an interesting the article "Brain Computation Is Organized via Power-of-Two-Based Permutation Logic" by Kun Xie et al in Frontiers in Systems Neuroscience (see this).
The proposed model is about how brain classifies neuronal inputs and suggests that the classification is based on Boolean algebra represents as subsets of n-element set for n inputs. The following represents my attempt to understand the model of the article.
The Mn=2n-1 (Mersenne number) labelling the algorithm is more than familiar to me.
- One can consider a situation in which one has n inputs identifiable as bits: bit could correspond to neuron firing or not. The question is however to classify various input combinations. The obvious criterion is how many bits are equal to 1 (corresponding neuron fires). The input combinations in the same class have same number of firing neurons and the number of subsets with k elements is given by the binomial coefficient B(n,k)= n!/k!(n-k)!. There are clearly n-1 different classes in the classification since no neurons firing is not a possible observation. The conceptualization would tell how many neurons fire but would not specify which of them.
- To represent these bit combinations one needs 2n-1 neuron groups acting as unit representing one particular firing combination. These subsets with k elements would be mapped to neuron cliques with k firing neutrons. For given input individual firing neurons (k=1) would represent features, lowest level information. The n cliques with k=2 neurons would represent a more general classification of input. One obtains Mn=2n-1 combinations of firing neurons since the situations in which no neurons are firing is not counted as an input.
- If all neurons are firing then all the however level cliques are also activated. Set theoretically the subsets of set partially ordered by the number of elements form an inclusion hierarchy, which in Boolean algebra corresponds to the hierarchy of implications in opposite direction. The clique with all neurons firing correspond to the most general statement implying all the lower level statements. At k:th level of hierarchy the statements are inconsistent so that one has B(n,k) disjoint classes.
I have made attempts to understand why Mersenne primes Mn and more generally primes near powers of 2 seem to be so important physically in TGD Universe.
- For instance, electron's p-adic prime corresponds to Mersenne prime M127 =2127-1, the largest not completely super-astrophysical Mersenne prime for which the mass of particle would be extremely small. Hadron physics corresponds to M107 and M89 to weak bosons and possible scaled up variant of hadron physics with mass scale scaled up by a factor 512 (=2(107-89)/2). Also Gaussian Mersennes seem to be physically important: for instance, muon and also nuclear physics corresponds to MG,n= (1+i)n-1, n=113.
- In biology the Mersenne prime M7= 27-1 is especially interesting. The number of statements in Boolean algebra of 7 bits is 128 and the number of statements that are consistent with given atomic statement (one bit fixed) is 26= 64. This is the number of genetic codons which suggests that the letters of code represent 2 bits. As a matter of fact, the so called Combinatorial Hierarchy M(n)= MM(n-1) consists of Mersenne primes n=3,7,127, 2127-1 and would have an interpretation as a hierarchy of statements about statements about ... It is now known whether the hierarchy continues beyond M127 and what it means if it does not continue. One can ask whether M127 defines a higher level code - memetic code as I have called it - and realizable in terms of DNA codon sequences of 21 codons (see this).
- The Gaussian Mersennes MG,n n=151,157,163,167, can be regarded as a number theoretical miracles since the these primes are so near to each other. They correspond to p-adic length scales varying between cell membrane thickness 10 nm and cell nucleus size 2.5 μm and should be of fundamental importance in biology. I have proposed that p-adically scaled down variants of hadron physics and perhaps also weak interaction physics are associated with them.
One can of course ask whether primes n defining Mersenne primes (see this) could define preferred numbers of inputs for subsystems of neurons. This would predict n=2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257,.. define favoured numbers of inputs. n=127 would correspond to memetic code.
- The states formed from n fermions form a Boolean algebra with 2n elements, but one of the elements is vacuum state and could be argued to be non-realizable. Hence Mersenne number Mn=2n-1. The realization as algebra of subsets contains empty set, which is also physically non-realizable. Mersenne primes are especially interesting as sine the reduction of statements to prime nearest to Mn corresponds to the number Mn-1 of physically representable Boolean
- Quantum information theory suggests itself as explanation for the importance of Mersenne primes since Mn would correspond the number of physically representable Boolean statements of a Boolean algebra with n-elements. The prime p≤ Mn could represent the number of elements of Boolean algebra representable p-adically (see this).
- In TGD Fermion Fock states basis has interpretation as elements of quantum Boolean algebra and fermionic zero energy states in ZEO expressible as superpositions of pairs of states with same net fermion numbers can be interpreted as logical implications. WCW spinor structure would define quantum Boolean logic as "square root of Kähler geometry". This Boolean algebra would be infinite-dimensional and the above classification for the abstractness of concept by the number of elements in subset would correspond to similar classification by fermion number. One could say that bosonic degrees of freedom (the geometry of 3-surfaces) represent sensory world and spinor structure (many-fermion states) represent that logical thought in quantum sense.
- Fermion number conservation would seem to represent an obstacle but in ZEO it can circumvented since zero energy states can be superpositions of pair of states with opposite fermion number F at opposite boundaries of causal diamond (CD) in such a manner that F varies. In state function reduction however localization to single value of F is expected to happen usually. If superconductors carry coherent states of Cooper pairs, fermion number for them is ill defined and this makes sense in ZEO but not in standard ontology unless one gives up the super-selection rule that fermion number of quantum states is well-defined.
See the article Why Mersenne Primes Are So Special? or the chapter Unified Number Theoretical Vision.