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

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242 11 Neural Models of Bayesian Belief Propagation Rajesh P. N. Rao time step, one can relax the equality in equation (11.21) and make log P (I ′ |θt i ) ∝ F(θi)I(t) for some linear filter F(θi) = ɛwi. This avoids the problem of calculating the normalization factor for P (I ′ |θt i ), which can be especially hard when I′ takes on continuous values such as in an image. A more challenging problem is to pick recurrent weights Uij such that equation (11.22) holds true. For equation (11.22) to hold true, we need to approximate a log-sum with a sum-of-logs. One approach is to generate a set of random probabilities xj(t) for t = 1, . . . , T and find a set of weights Uij that satisfy: j Uij log xj(t) ≈ log P (θ t i|θ t−1 j )xj(t) j (11.24) for all i and t. This can be done by minimizing the squared error in equation (11.24) with respect to the recurrent weights Uij. This empirical approach, followed in [30], is used in some of the experiments below. An alternative approach is to exploit the nonlinear properties of dendrites as suggested in the following section. 11.3.2 Exact Inference in Nonlinear Networks A firing rate model that takes into account some of the effects of nonlinear filtering in dendrites can be obtained by generalizing equation (11.18) as follows: vi(t + 1) = f wiI(t) + g Uijvj(t) , (11.25) j where f and g model nonlinear dendritic filtering functions for feedforward and recurrent inputs. By comparing this equation with the belief propagation equation in the log domain (equation (11.19)), it can be seen that the first equation can implement the second if: vi(t + 1) = log m t i (11.26) f wiI(t) = log P (I ′ |θ t i) (11.27) g Uijvj(t) = log P (θ t i|θ t−1 j )m t−1,t j (11.28) j j In this model (figure 11.2B), N neurons represent log mt i (i = 1, . . . , N) in their firing rates. The dendritic filtering functions f and g approximate the logarithm function, the feedforward weights wi act as a linear filter on the input to yield the likelihood P (I ′ |θt i ) and the recurrent synaptic weights Uij directly encode the transition probabilities P (θt i |θt−1 j ). The normalization step is com- puted as in equation (11.23) using a separate group of neurons that represent log posterior probabilities log m t,t+1 i and that convey these probabilities for use . in equation (11.28) by the neurons computing log m t+1 i
11.3 Neural Implementations of Belief Propagation 243 11.3.3 Inference Using Noisy Spiking Neurons Spiking Neuron Model The models above were based on firing rates of neurons, but a slight modification allows an interpretation in terms of noisy spiking neurons. Consider a variant of equation (11.16) where v represents the membrane potential values of neurons rather than their firing rates. We then obtain the classic equation describing the dynamics of the membrane potential vi of neuron i in a recurrent network of leaky integrate-and-fire neurons: τ dvi dt = −vi + wijIj + uijv ′ j, (11.29) j j where τ is the membrane time constant, Ij denotes the synaptic current due to input neuron j, wij represents the strength of the synapse from input j to recurrent neuron i, v ′ j denotes the synaptic current due to recurrent neuron j, and uij represents the corresponding synaptic strength. If vi crosses a threshold T , the neuron fires a spike and vi is reset to the potential vreset. Equation (11.29) can be rewritten in discrete form as: vi(t + 1) = vi(t) + ɛ(−vi(t) + wijIj(t)) + uijv ′ j(t)) (11.30) i.e. vi(t + 1) = ɛ wijIj(t) + Uijv ′ j(t), (11.31) j where ɛ is the integration rate, Uii = 1 + ɛ(uii − 1) and for i = j, Uij = ɛuij. The nonlinear variant of the above equation that includes dendritic filtering of input currents in the dynamics of the membrane potential is given by: vi(t + 1) = f wijIj(t) + g Uijv ′ j(t) , (11.32) j j where f and g are nonlinear dendritic filtering functions for feedforward and recurrent inputs. We can model the effects of background inputs and the random openings of membrane channels by adding a Gaussian white noise term to the right-hand side of equations (11.31) and (11.32). This makes the spiking of neurons in the recurrent network stochastic. Plesser and Gerstner [27] and Gerstner [11] have shown that under reasonable assumptions, the probability of spiking in such noisy neurons can be approximated by an “escape function” (or hazard function) that depends only on the distance between the (noise-free) membrane potential vi and the threshold T . Several different escape functions were found to yield similar results. We use the following exponential function suggested in [11] for noisy integrate-and-fire networks: j P (neuron i spikes at time t) = ke (vi(t)−T ) , (11.33) where k is an arbitrary constant. We use a model that combines equations (11.32) and (11.33) to generate spikes. j j
Page 1 and 2: 11 Neural Models of Bayesian Belief
Page 3 and 4: 11.2 Bayesian Inference through Bel
Page 5 and 6: 11.2 Bayesian Inference through Bel
Page 7: 11.3 Neural Implementations of Beli
Page 11 and 12: 11.4 Results 245 deviation σ. Figu
Page 13 and 14: 11.4 Results 247 Nonlinear Spiking
Page 15 and 16: 11.4 Results 249 again leaky-integr
Page 17 and 18: 11.4 Results 251 a high value for P
Page 19 and 20: 11.5 Discussion 253 an evidence acc
Page 21 and 22: 11.5 Discussion 255 the neural popu
Page 23 and 24: 11.5 Discussion 257 of various ioni
Page 25 and 26: 11.5 Discussion 259 [28] Pouget A,
Page 27 and 28: 11.5 Discussion 261 50% 100% D 0.05
Page 29 and 30: 11.5 Discussion 263 Normalized Firi
Page 31 and 32: Index acausal, 122 attention, 101 a
Page 33: Index 319 regularization, 11 revers

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