Probabilistic population codes for Bayesian decision making.

Beck, J. M., Ma, W. J., Kiani, R., Hanks, T., Churchland, A. K., Roitman, J., Shadlen, M. N., Latham, P. E., and Pouget, A.
Neuron, 60:1142–1152, 2008
DOI, Google Scholar

Abstract

When making a decision, one must first accumulate evidence, often over time, and then select the appropriate action. Here, we present a neural model of decision making that can perform both evidence accumulation and action selection optimally. More specifically, we show that, given a Poisson-like distribution of spike counts, biological neural networks can accumulate evidence without loss of information through linear integration of neural activity and can select the most likely action through attractor dynamics. This holds for arbitrary correlations, any tuning curves, continuous and discrete variables, and sensory evidence whose reliability varies over time. Our model predicts that the neurons in the lateral intraparietal cortex involved in evidence accumulation encode, on every trial, a probability distribution which predicts the animal’s performance. We present experimental evidence consistent with this prediction and discuss other predictions applicable to more general settings.

Review

In this article the authors apply probabilistic population coding as presented in Ma et al. (2006) to perceptual decision making. In particular, they suggest a hierarchical network with a MT and LIP layer in which the firing rates of MT neurons encode the current evidence for a stimulus while the firing rates of LIP neurons encode the evidence accumulated over time. Under the made assumptions it turns out that the accumulated evidence is independent of nuisance parameters of the stimuli (when they can be interpreted as contrasts) and that LIP neurons only need to sum (integrate) the activity of MT neurons in order to represent the correct posterior of the stimulus given the history of evidence. They also suggest a readout layer implementing a line attractor which reads out the maximum of the posterior under some conditions.

Details

Probabilistic population coding is based on the definition of the likelihood of stimulus features p(r|s,c) as an exponential family distribution of firing rates r. A crucial requirement for the central result of the paper (that LIP only needs to integrate the activity of MT) is that nuisance parameters c of the stimulus s do not occur in the exponential itself while the actual parameters of s only occur in the exponential. This restricts the exponential family distribution to the “Poisson-like family”, as they call it, which requires that the tuning curves of the neurons and their covariance are proportional to the nuisance parameters c (details for this need to be read up in Ma et al., 2006). The point is that this is the case when c corresponds to contrast, or gain, of the stimulus. For the considered random dot stimuli the coherence of the dots may indeed be interpreted as the contrast of the motion in the sense that I can imagine that the tuning curves of the MT neurons are multiplicatively related to the coherence of the dots.

The probabilistic model of the network activities is setup such that the firing of neurons in the network is an indirect, noisy observation of the underlying stimulus, but what we are really interested in is the posterior of the stimulus. So the question is how you can estimate this posterior from the network firing rates. The trick is that under the Poisson-like distribution the likelihood and posterior share the same exponential such that the posterior becomes proportional to this exponential, because the other parts of the likelihood do not depend on the stimulus s (they assume a flat prior of s such that you don’t need to consider it when computing the posterior). Thus, the probability of firing in the network is determined from the likelihood while the resulting firing rates simultaneously encode the posterior. Mind-boggling. The main contribution from the authors then is to show, assuming that firing rates of MT neurons are driven from the stimulus via the corresponding Poisson-like likelihood, that LIP neurons only need to integrate the spikes of MT neurons in order to correctly represent the posterior of the stimulus given all previous evidence (firing of MT neurons). Notice, that they also assume that LIP neurons have the same tuning curves with respect to the stimulus as MT neurons and that the neurons in LIP sum the activity of this MT neuron which they share a tuning curve with. They note that a naive procedure like that, i.e. a single neuron integrating MT firing over time, would quickly saturate its activity. So they show, and that is really cool, that global inhibition in the LIP network does not affect the representation of the posterior, allowing them to prevent saturation of firing while maintaining the probabilistic interpretation.

So far to the theory. In practice, i.e. experiments, the authors do something entirely different, because “these results are important, but they are based on assumptions that are not necessarily exactly true in vivo. […] It is therefore essential that we test our theory in biologically realistic networks.” Now, this is a noble aim, but what exactly do we learn about this theory, if all results are obtained using methods which violate the assumptions of the theory? For example, neither the probability of firing in MT nor LIP is Poisson-like, LIP neurons not just integrate MT activity, but are also recurrently connected, LIP neurons have local inhibition (they are leaky integrators, inhibition between LIP neurons depending on tuning properties) instead of global inhibition and LIP neurons have an otherwise completely unmotivated “urgency signal” whose contribution increases with time (this stems from experimental observations). Without any concrete links between the two models in theory (I guess, the main ideas are similar, but the details are very different) it has to be shown that they are similar using experimental results. In any case, it is hard to differentiate between contributions from the probabilistic theory and the network implementation, i.e., how much of the fit between experimental findings in monkeys and the behaviour of the model is due to the chosen implementation and how much is due to the probabilistic interpretation?

Results

The overall aim of the experiments / simulations in the paper is to show that the proposed probabilistic interpretation is compatible with the experimental findings in monkey LIP. The hypothesis is that LIP neurons encode the posterior of the stimulus as suggested in the theory. This hypothesis is false from the start, because some assumptions of the theory apparently don’t apply to neurons (as acknowledged by the authors). So the new hypothesis is that LIP neurons approximately encode some posterior of the stimulus. The requirement for this posterior is that updates of the posterior should take the uncertainty of the evidence and the uncertainty of the previous estimate of the posterior into account which the authors measure as a linear increase of the log odds of making a correct choice, log[ p(correct) / (1-p(correct)) ], with time together with the dependence of the slope of this linear increase on the coherence (contrast) of the stimulus. I did not follow up why the previous requirement is related to the log odds in this way, but it sounds ok. Remains the question how to estimate the log odds from simulated and real neurons. For the simulated neurons the authors approximate the likelihood with a Poisson-like distribution whose kernel (parameters) were estimated from the simulated firing rates. They argue that it is a good approximation, because linear estimates of the Fisher information appear to be sufficient (I can’t comment on the validity of this argument). A similar approximation of the posterior cannot be done for real LIP neurons, because of a lack of multi-unit recordings which estimate the response of the whole LIP population. Instead, the authors approximate the log odds from measured firing rates of neurons tuned to motion in direction 0 and 180 degrees via a linear regression approach described in the supplemental data.

The authors show that the log-odds computed from the simulated network exhibit the desired properties, i.e., the log-odds linearly increase with time (although there’s a kink at 50ms which supposedly is due to the discretisation) and depend on the coherence of the motion such that the slope of the log-odds increases also when coherence is increased within a trial. The corresponding log-odds of real LIP neurons are far noisier and, thus, do not allow to make definite judgements about linearity. Also, we don’t know whether their slopes would actually change after a change in motion coherence during a trial, as this was never tested (it’s likely, though).

In order to test whether the proposed line attractor network is sufficient to read out the maximum of the posterior in all conditions (readout time and motion coherence) the authors compare a single (global) readout with local readouts adapted for a particular pair of readout time and motion coherence. However, the authors don’t actually use attractor networks in these experiments, but note that these are equivalent to local linear estimators and so use these. Instead of comparing the readouts from these estimators with the actual maximum of the posterior, they only compare the variance of the estimators (Fisher information) which they show to be roughly the same for the local and global estimators. From this they conclude that a single, global attractor network could read out the maximum of the (approximated) posterior. However, this is only true, if there’s no additional bias of the global estimator which we cannot see from these results.

In an additional analysis the authors show that the model qualitatively replicates the behavioural variables (probability correct and reaction time). However, these are determined from the LIP activities in a surprisingly ad-hoc way: the decision time is determined as the time when any one of the simulated LIP neurons reaches a threshold defined on the probability of firing and the decision is determined as the preferred direction of the neuron hitting the threshold (for 2 and 4 choice tasks the response is determined as the quadrant of the motion direction in which the preferred direction of the neuron falls). Why do the authors not use the attractor network to readout the response here? Also, the authors use a lower threshold for the 4-choice task than for the 2-choice task. This is strange, because one of the main findings of the Churchland et al. (2008) paper was that the decision in both, 2- and 4-choice tasks, appears to be determined by a common decision threshold while the initial firing rates of LIP neurons were lower for 4-choice tasks. Here, they also initialise with lower firing rates in the 4-choice task, but additionally choose a lower threshold. They don’t motivate this. Maybe it was necessary to fit the data from Churchland et al. (2008). This discrepancy between data and model is even more striking as the authors of the two papers partially overlap. So, do they deem the corresponding findings of Churchland et al. (2008) not important enough to be modelled, is it impossible to be modelled within their framework, or did they simply forget?

Finally, also the build-up rates of LIP neurons seem to be qualitatively similar in the simulation and the data, although they are consistently lower in the model. The build-up rates for the model are estimated from the first 50ms within each trial. However, the log-odds ratio had this kink at 50ms after which its slope was larger. So, if this effect is also seen directly in the firing rates, the fit of the build-up rates to the data may even be better, if probability of firing after 50ms is used. In Fig. 2C no such kink can be seen in the firing rates, but this is only data for 2 neurons in the model.

Conclusion

Overall the paper is very interesting and stimulating. It is well written and full of sound theoretical results which originate from previous work of the authors. Unfortunately, biological nature does not completely fit the beautiful theory. Consequently, the authors run experiments with more plausible neural networks which only approximately implement the theory. So what conclusions can we draw from the presented results? As long as the firing of MT neurons reflects the likelihood of a stimulus (their MT network is setup in this way), probably a wide range of networks which accumulate this firing will show responses similar to real LIP neurons. It is not easy to say whether this is a consequence of the theory, which states that MT firing rates should be simply summed over time in order to get the right posterior, because of the violation of the assumptions of the theory in more realistic networks. It could also be that more complicated forms of accumulation are necessary such that LIP firing represents the correct posterior. Simple summing then just represents a simple approximation. Also, I don’t believe that the presented results can rule out the possibility of sampling based coding of probabilities (see Fiser et al., 2010) for decision making as long as also the sampling approach would implement some kind of accumulation procedure (think of particle filters – the implementation in a recurrent neural network would probably be quite similar).

Nevertheless, the main point of the paper is that the activity in LIP represents the full posterior and not only MAP estimates or log-odds. Consequently, the model very easily extends to the case of continuous directions of motion which is in contrast to previous, e.g., attractor-based, neural models. I like this idea. However, I cannot determine from the experiments whether their network actually implements the correct posterior, because all their tests yield only indirect measures based on approximated analyses. Even so, it is pretty much impossible to verify that the firing of LIP neurons fits to the simulated results as long as we cannot measure firing of a large part of the corresponding neural population in LIP.

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