Bayesian Brain - Chapter 11 Neural Models of Bayesian Belief Propagation Rajesh P.N. Rao Summary by B.-H. Kim Biointelligence Lab School of Computer Sci. & Eng. Seoul National University

(c) SNU CSE Biointelligence Lab2 Outline Cortical neuron Computing P(preferred state | current/past inputs) Analogy Spiking prob. ∝ P(S|D) Bayesian Inference Neurophysiology Input from inhibitory neurons Input from excitatory neurons Transition prob. btw states Prob. normalization Feedback from higher to lower areas Prior probabilities Models for probabilistic computation in networks of neuron-like elements

Introduction (c) SNU CSE Biointelligence Lab3 Inference over time using hidden Markov model (HMM) [A] Inference in a hierarchical graphical model [A] Visual motion detection and decision-making Understanding attentional effects in the primate visual cortex Models for neural implementation of the belief propagation algorithm For Bayesian inference [B] model application

Bayesian Inference through Belief Propagation (c) SNU CSE Biointelligence Lab4 Sum over many r.v.  exponential growth of comp. time Belief propagation (local operations) P(R) P(M) mCRmCR Efficient computation of the posterior probabilites

Belief Propagation over Time In HMM (hidden Markov model) (c) SNU CSE Biointelligence Lab5 Emission probabilities Message (forward)

Hierarchical Belief Propagation An example of 3-level graphical model for images (c) SNU CSE Biointelligence Lab6 Messages Posterior prob.

Belief Propagation over Time – Approximate Inference in Linear Recurrent Networks Linear recurrent network with firing dynamics  Commonly used neural architecture for modeling cortical response properties  Discrete form 7 I v (output firing rate) W (forward weight matrix) U (recurrent weight matrix) (11.5)

Belief Propagation over Time – Exact Inference in Nonlinear Networks Firing rate model that takes into account some of the effects of nonlinear filtering in dendrites 8 I v W (forward weight matrix) U (recurrent weight matrix) (linear recurrent network) (f, g: nonlinear dendritic filtering functions)

Neural Circuits (c) SNU CSE Biointelligence Lab9

Results Example 1: Detecting Visual Motion A prominent property of visual cortical cells in area (e.g. V1, MT) is selectivity to the direction of visual motion Interpretation on the activity of these cells  the posterior probability of stimulus motion in a particular direction  Given a series of input images Experiment  1D motion in an image with two possible motion directions: L or R (c) SNU CSE Biointelligence Lab10

Visual Cortex in Brains of Primates (c) SNU CSE Biointelligence Lab11

Results Example 1: Detecting Visual Motion (NIPS 2005) (c) SNU CSE Biointelligence Lab12

Results Example 2: Bayesian Decision-Making in a Random-Dots Task Dots motion discrimination task  Stimulus  An image sequence showing a group of moving dots  A fixed fraction of which are randomly selected at each frame and moved in a fixed direction (the rest are moved in random direction)  Coherence: the fraction of dots moving in the same direction  Task  Decide the direction of motion of the coherently moving dots  Data  Phychophysical performance of humans and monkeys + neural responses in brain areas such as MT and LIP Goal of the experiment  Explore the extent to which the proposed models for neural belief propagation can explain the exisiting data (c) SNU CSE Biointelligence Lab13

Results Example 2: Bayesian Decision-Making in a Random-Dots Task (c) SNU CSE Biointelligence Lab14

Hierarchical Belief Propagation - Noisy Spiking Neuron Model v represents the membrane potential values of neurons rather than their firing rates Recurrent network of leaky integrate-and-fire neurons  If v i crosses a threshold T, the neuron fires a spike and v i is reset to the potential v reset  Discrete form  Nonlinear variant (c) SNU CSE Biointelligence Lab15 background inputs Random openings of membrane channel Gaussian white noise  Escape function

Results Example 3: Attention in the Visual Cortex The responses modulation of neurons incortical areas V2 and V4 by attention to particular location within an input image (c) SNU CSE Biointelligence Lab16 Multiplicative modulation due to attention Input image configuratoin and conditional probabilities

Effects of Attention on Responses in the Presence of Distractors (c) SNU CSE Biointelligence Lab17

Effects of Attention on Neighboring Spatial Locations (c) SNU CSE Biointelligence Lab18

Related Models Models based on log-likelihood ratios Inference using distributional codes Hierarchical inference (c) SNU CSE Biointelligence Lab19

Open Problems and Future Challenges Learning and adaptation The use of spikes in prob. Representations How the dendritic nonlinearities could be exploited to implement belief propagation Exploring graphical models that are inspired by neurobiology (c) SNU CSE Biointelligence Lab20