Neural Models of Bayesian Belief Propagation Rajesh ... - Washington

244 11 Neural Models of Bayesian Belief Propagation Rajesh P. N. Rao
Inference in Spiking Networks
By comparing the membrane potential equation (11.32) with the belief propagation
equation in the log domain (equation (11.19)), we can postulate the
following correspondences:
vi(t + 1) = log m t i (11.34)
f
wijIj(t) = log P (I ′ |θ t i) (11.35)
j
g
j
Uijv ′ j(t) = log
j
P (θ t i|θ t−1
j )m t−1,t
j
(11.36)
The dendritic filtering functions f and g approximate the logarithm function,
the synaptic currents Ij(t) and v ′ j (t) are approximated by the corresponding
instantaneous firing rates, and the recurrent synaptic weights Uij encode the
).
transition probabilities P (θt i |θt−1 j
Since the membrane potential vi(t+1) is assumed to be equal to log mt i (equation
(11.34)), we can use equation (11.33) to calculate the probability of spiking
for each neuron i as:
(vi(t+1)−T )
P (neuron i spikes at time t + 1) ∝ e
∝ e
(log mti
−T )
(11.37)
(11.38)
∝ m t i (11.39)
Thus, the probability of spiking (or, equivalently, the instantaneous firing rate)
for neuron i in the recurrent network is directly proportional to the message m t i ,
which is the posterior probability of the neuron’s preferred state and current
input given past inputs. Similarly, the instantaneous firing rates of the group of
neurons representing log m t,t+1
i is proportional to m t,t+1
i , which is the precisely
the input required by equation (11.36).
11.4 Results
11.4.1 Example 1: Detecting Visual Motion
We first illustrate the application of the linear firing rate-based model (section
11.3.1) to the problem of detecting visual motion. A prominent property
of visual cortical cells in areas such as V1 and MT is selectivity to the direction
of visual motion. We show how the activity of such cells can be interpreted as
representing the posterior probability of stimulus motion in a particular direction,
given a series of input images. For simplicity, we focus on the case of 1D
motion in an image consisting of X pixels with two possible motion directions:
leftward (L) or rightward (R).
Let the state θij represent a motion direction j ∈ {L, R} at spatial location
i. Consider a network of N neurons, each representing a particular state
θij (figure 11.4A). The feedforward weights are assumed to be Gaussians, i.e.
F(θiR) = F(θiL) = F(θi) = Gaussian centered at location i with a standard