Reciprocally Coupled Local Estimators Implement Bayesian Information Integration Distributively
Psychophysical experiments have demonstrated that the brain integrates information from multiple sensory cues in a near Bayesian optimal manner. The present study proposes a novel mechanism to achieve this. We consider two reciprocally connected networks, mimicking the integration of heading direction information between the dorsal medial superior temporal (MSTd) and the ventral intraparietal (VIP) areas. Each network serves as a local estimator and receives an independent cue, either the visual or the vestibular, as direct input for the external stimulus. We find that positive reciprocal interactions can improve the decoding accuracy of each individual network as if it implements Bayesian inference from two cues. Our model successfully explains the experimental finding that both MSTd and VIP achieve Bayesian multisensory integration, though each of them only receives a single cue as direct external input. Our result suggests that the brain may implement optimal information integration distributively at each local estimator through the reciprocal connections between cortical regions.
"Reciprocally Coupled Local Estimators Implement 
      Bayesian Information Integration Distributively" puts forward a new take 
      on Bayesian integration of multimodal cues. Instead of assuming a special 
      area in the brain, where evidence from various sensory cues is combined 
      (as in Ma and all, 2006), the authors consider a scenario, whereby each 
      area receiving direct afferent input from a single modality (i.e. visual) 
      combines information from other modalities (e.g. vestibular) via 
      reciprocal connections. In the example analysed by the authors, and 
      under a number of suitable assumptions, the cue integration they observe 
      in their networks is close to Bayes-optimal. Building up on work 
      of Fung and all (2010), the authors derive theoretical predictions for the 
      integration of information in reciprocally coupled ring attractors 
      (CANNs), which they also confirm by simulations. The reader is led through 
      the general steps of the analysis, while details are provided in the 
      supplementary material. The main result is that for a sufficiently 
      weak input, and sufficiently narrow interactions (HWHH = 67 degrees for 
      their example of heading direction), each of the networks combines 
      information from the two cues in a Bayes optimal manner. That is, the 
      decoding precision of a single network when both cues are available is a 
      sum of this network's precisions when only a single cue is present, i.e. 
      p(theta| c1, c2) = p1(theta|c1) + p1(theta|c2), where c1 is either 
      - the directly encoded cue, e.g. visual, which has a higher decoding 
      accuracy, or - the other modality cue, reported via reciprocal 
      connections, with decoding accuracy growing with the strength of 
      reciprocal connections. This result is interesting despite the 
      simplifying assumptions. My only major concern would be the lack of 
      attention given to the posterior mean. As indicated by Fig. 3C, if the 
      cues were not congruent, the networks would display bias towards the mean 
      of the two cues larger than predicted by Bayesian inference. Although 
      presented, this effect is not discussed anywhere in the article. Likewise, 
      the Bayes estimate of the mean could be plotted together with the results 
      of the simulations in Fig. 4. Quality This article is 
      technically sound, supported well by theoretical predictions and 
      simulations. It would be possible to evaluate weaknesses of the work in 
      more detail (see above). Clarity The article is clearly 
      written, although it would benefit from one more careful reading. In 
      Fig. 4 legend, values of mu_1 and mu_2 seem to be swapped. L. 395, 
      reference to Eq. 3 was possibly meant to be Eq. 2. Originality 
      To the best of my knowledge, the approach is new. The work it builds 
      upon is well referenced. Significance As above, I believe the 
      impact of this work will reach a wide audience, far wider than modellers 
      working with attractor networks.