Bernal S D_2010.pdf - University of Plymouth

3.4. EXISTING MODELS
3.4.1.3 Local inference circuits model
Concurrent lo the publication of the previous model, I jtvak and Ullman (2009) published an
alternative implementation of belief propagation using spiking neurons. The latter employs pop­
ulations of standard leaky integrale-und-lire neurons to implement the belief revision algorithm
in pairwise Markov random fields. Belief revision is analogous to belief propagation except
that it replaces the sum operation with the max operation, i.e. uses the max-product instead of
the sum-product algorithm, thus obtaining the maximum-a-posleriori estimate (also called the
most probable explanation) instead of the posterior marginal probability. Additionally, the algo­
rithm is implemented in the log domain, which leads to a linal neuronal implementation based
on a max-sum scheme, called belief consolidalion. Pairwise Markov random lields are a type
of undirected graphical model, which share many properties with directed graphical models
(Bayesian networks), but are not inlfrchangeable (see Section 3.J.3).
To implement the belief consolidalion algorithm, the model uses building blocks called local
inference circuit.^ (MNCs). Fach neuronal LINC is connected to other LINCs according to the
graph structure, and propagates the same message to all neighbours. Each LINC roughly im­
plements the operations peri'ormed locally by each node in the graph using smaller elementary
circuits thai approximate the two mathematical operations: a linear summation circuit and a
maximization circuit. The model uses populations of leaky integrate-and-fire neurons to im-
plemenl these computations. The synaptic weights between the different elementary circuits
define their specific functional properties. The mean rate of ihe neural populations during short
peritKls of lime (few tens of milliseconds), represents the values of messages computed during
the inference process,
Hach neuron.il I.INC uses A' (number of neighbours) x S (number of states) weighted maxi­
mization circuits, which compute the maximum value for each slate of the inpul nodes. Before
finding the maximum value, the circuit uses a linear summation element to add the correspond­
ing weight to each input message (in Ihe log domain, weights are additive). The weighted
maximum results for each state are then combined in the N corresponding summation circuits.
The vector of single valued outputs of each summation circuit represents the output message of
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