We study the dynamics of a simple message-passing decoder for LDGM channel coding by using the generating functional analysis (GFA). The decoder addressed here is one of the simplest examples, which is characterized by a sparse random graph with many short loops. The GFA allows us to study the dynamics of iterative systems in an exact way in the large codeword length limit.

We analyse the effect of chemical neuro-modulation on collective processes in Ising spin neural networks with separable Hebbian type synaptic interactions. Neuro-modulation is taken into account in the most simple way: a modulator-specific subset of neurons is prevented from transmitting signals. However, the presence of neuro-modulators is taken into account also during the learning stage, which leads to non-symmetric interaction matrices. We derive (in the limit of an infinite system size) the macroscopic laws that determine the system’s evolution in time on the level of order parameters. These laws are very transparant and show that, within the proposed framework, one can understand the functioning of neuro-modulators as follows: their role is to choose from the repertoire of learned behaviour a particular mode of operation. By considering specific examples of learning stages we indicate how neuro-modulation might be used by the brain as an extra degree of freedom for (a) performing selective pattern reconstruction, (b) controlling the reproduction speed of stored pattern sequences or (c) for choosing a particular path from a set of partially overlapping stored trajectories through state space (at points where the trajectories separate).