Tag: hierarchy

Decision-related activity in sensory neurons reflects more than a neuron's causal effect.

Nienborg, H. and Cumming, B. G.
Nature, 459:89–92, 2009
DOI, Google Scholar

Abstract

During perceptual decisions, the activity of sensory neurons correlates with a subject’s percept, even when the physical stimulus is identical. The origin of this correlation is unknown. Current theory proposes a causal effect of noise in sensory neurons on perceptual decisions, but the correlation could result from different brain states associated with the perceptual choice (a top-down explanation). These two schemes have very different implications for the role of sensory neurons in forming decisions. Here we use white-noise analysis to measure tuning functions of V2 neurons associated with choice and simultaneously measure how the variation in the stimulus affects the subjects’ (two macaques) perceptual decisions. In causal models, stronger effects of the stimulus upon decisions, mediated by sensory neurons, are associated with stronger choice-related activity. However, we find that over the time course of the trial these measures change in different directions-at odds with causal models. An analysis of the effect of reward size also supports this conclusion. Finally, we find that choice is associated with changes in neuronal gain that are incompatible with causal models. All three results are readily explained if choice is associated with changes in neuronal gain caused by top-down phenomena that closely resemble attention. We conclude that top-down processes contribute to choice-related activity. Thus, even forming simple sensory decisions involves complex interactions between cognitive processes and sensory neurons.

Review

They investigated the source of the choice probability of early sensory neurons. Choice probability quantifies the difference in firing rate distributions separated by the behavioural response of the subject. The less overlap between the firing rate distributions for one response and its alternative (in two-choice tasks), the greater the choice probability. Importantly, they restricted their analysis to trials in which the stimulus was effectively random. In random dot motion experiments this corresponds to 0% coherent motion, but here they used a disparity discrimination task and looked at disparity selective neurons in macaque area V2. The mean contribution from the stimulus, therefore, should have been 0. Yet, they found that choice probability was above 0.5 indicating that the firing of the neurons still could predict the final response, but why? They consider two possibilities: 1) the particular noise in firing rates of sensory neurons causes, at least partially, the final choice. 2) The firing rate of sensory neurons reflects choice-related effects induced by top-down influences from more decision-related areas.

Note that the choice probability they use is somewhat corrected for influences from the stimulus by considering the firing rate of a neuron in response to a particular disparity, but without taking choices into account. This correction reduced choice probabilities a bit. Nevertheless, they remained significantly above 0.5. This result indicates that the firing rate distributions of the recorded neurons were only little affected by which disparities were shown in individual frames when these distributions are defined depending on the final choice. I don’t find this surprising, because there was no consistent stimulus to detect from the random disparities and the behavioural choices were effectively random.

Yet, the particular disparities presented in individual trials had an influence on the final choice. They used psychophysical reverse correlation to determine this. The analysis suggests that the very first frames had a very small effect which is followed by a steep rise in influence of frames at the beginning of a trial (until about 200ms) and then a steady decline. This result can mean different things depending on whether you believe that evidence accumulation stops once you have reached a threshold, or whether evidence accumulation continues until you are required to make a response. Shadlen is probably a proponent of the first proposition. Then, the decreasing influence of the stimulus on the choice just reflects the smaller number of trials in which the threshold hasn’t been reached, yet. Based on the second proposition, the result means that the weight of individual pieces of evidence during accumulation reduces as you come closer to the response. Currently, I can’t think of decisive evidence for either proposition, but it has been shown in perturbation experiments that stimulus perturbations close to a decision, late in a trial had smaller effects on final choices than perturbations early in a trial (Huk and Shadlen, 2005).

Back to the source of above chance-level choice probabilities. The authors argue, given the decreasing influence of the stimulus on the final choice and assuming that the influence of the stimulus on sensory neurons stays constant, that choice probabilities should also decrease towards the end of a trial. However, choice probabilities stay roughly constant after an initial rise. Consequently, they infer that the firing of the neurons must be influenced from other sources, apart from the stimulus, which are correlated with the choice. They consider two of these sources: i) Lateral, sensory neurons which could reflect the final decision better. ii) Higher, decision related areas which, for example, project a kind of bias onto the sensory neurons. The authors strongly prefer ii), also because they found that the firing of sensory neurons appears to be gain modulated when contrasting firing rates between final choices. In particular, firing rates showed a larger gain (steeper disparity tuning curve of neuron) when trials were considered which ended with the behavioural choice corresponding to the preferred dispartiy of the neuron. In other words, the output of a neuron was selectively increased, if that neuron preferred the disparity which was finally chosen. Equivalently, the output of a neuron was selectively decreased, if that neuron preferred a different disparity than the one which was finally chosen. This gain difference explains at least part of the difference in firing rate distributions which the choice probability measures.

They also show an interesting effect of reward size on the correlation between stimulus and final choice: Stimulus had larger influence on choice for larger reward. Again, if the choice probabilities were mainly driven by stimulus, bottom-up related effects and the stimulus had a larger influence on final choice in high reward trials, then choice probabilities should have been higher in high reward trials. The opposite was the case: choice probabilities were lower in high reward trials. The authors explain this using the previous bias hypothesis: The measured choice probabilities reflect something like an attentional gain or bias induced by higher-level decision-related areas. As the stimulus becomes more important, the bias looses influence. Hence, the choice probabilities reduce.

In summary, the authors present convincing evidence that already sensory neurons in early visual cortex (V2) receive top-down, decision-related influences. Compared with a previous paper (Nienborg and Cumming, 2006) the reported choice probabilities here were quite similar to those reported there, even though here only trials with complete random stimuli were considered. I would have guessed that choice probabilities would be considerably higher for trials with an actually presented stimulus. Why is there only a moderate difference? Perhaps there actually isn’t. My observation is only based on a brief look at the figures in the two papers.

Perceptions as hypotheses: saccades as experiments.

Friston, K., Adams, R. A., Perrinet, L., and Breakspear, M.
Front Psychol, 3:151, 2012
DOI, Google Scholar

Abstract

If perception corresponds to hypothesis testing (Gregory, 1980); then visual searches might be construed as experiments that generate sensory data. In this work, we explore the idea that saccadic eye movements are optimal experiments, in which data are gathered to test hypotheses or beliefs about how those data are caused. This provides a plausible model of visual search that can be motivated from the basic principles of self-organized behavior: namely, the imperative to minimize the entropy of hidden states of the world and their sensory consequences. This imperative is met if agents sample hidden states of the world efficiently. This efficient sampling of salient information can be derived in a fairly straightforward way, using approximate Bayesian inference and variational free-energy minimization. Simulations of the resulting active inference scheme reproduce sequential eye movements that are reminiscent of empirically observed saccades and provide some counterintuitive insights into the way that sensory evidence is accumulated or assimilated into beliefs about the world.

Review

In this paper Friston et al. introduce the notion that an agent (such as the brain) minimizes uncertainty about its state in the world by actively sampling those states which minimise the uncertainty of the agent’s posterior beliefs, when visited some time in the future. The presented ideas can also be seen as reply to the commonly formulated dark-room-critique of Friston’s free energy principle which states that under the free energy principle an agent would try to find a dark, stimulus-free room in which sensory input can be perfectly predicted. Here, I review these ideas together with the technical background (see also a related post about Friston et al., 2011). Although I find the presented theoretical argument very interesting and sound (and compatible with other proposals for the origin of autonomous behaviour), I do not think that the presented simulations conclusively show that the extended free energy principle as instantiated by the particular model chosen in the paper leads to the desired exploratory behaviour.

Introduction: free energy principle and the dark room

Friston’s free energy principle has gained considerable momentum in the field of cognitive neuroscience as a unifying framework under which many cognitive phenomena may be understood. Its main axiom is that an agent tries to minimise the long-term uncertainty about its state in the world by executing actions which make prediction of changes in the agent’s world more precise, i.e., which minimise surprises. In other words, the agent tries to maintain a sort of homeostasis with its environment.

While homeostasis is a concept which most people happily associate with bodily functions, it is harder to reconcile with cognitive functions which produce behaviour. Typically, the counter-argument for the free energy principle is the dark-room-problem: changes in a dark room can be perfectly predicted (= no changes), so shouldn’t we all just try to lock ourselves into dark rooms instead of frequently exploring our environment for new things?

The shortcoming of the dark-room-problem is that an agent cannot maintain homeostasis in a dark room, because, for example, its bodily functions will stop working properly after some time without water. There may be many more environmental factors which may disturb the agent’s dark room pleasure. An experienced agent knows this and has developed a corresponding model about its world which tells it that the state of its world becomes increasingly uncertain as long as the agent only samples a small fraction of the state space of the world, as it is the case when you are in a dark room and don’t notice what happens outside of the room.

The present paper formalises this idea. It assumes that an agent only observes a small part of the world in its local surroundings, but also maintains a more comprehensive model of its world. To decrease uncertainty about the global state of the world, the agent then explores other parts of the state space which it beliefs to be informative according to its current estimate of the global world state. In the remainder I will present the technical argument in more detail, discuss the supporting experiments and conclude with my opinion about the presented approach.

Review of theoretical argument

In previous publications Friston postulated that agents try to minimise the entropy of the world states which they encounter in their life and that this minimisation is equivalent to minimising the entropy of their sensory observations (by essentially assuming that the state-observation mapping is linear). The sensory entropy can be estimated by the average of sensory surprise (negative model evidence) across (a very long) time. So the argument goes that an agent should minimise sensory surprise at all times. Because sensory surprise cannot usually be computed directly, Friston suggests a variational approximation in which the posterior distribution over world states (posterior beliefs) and model parameters is separated. Further, the posterior distributions are approximated with Gaussian distributions (Laplace approximation). Then, minimisation of surprise is approximated by minimisation of Friston’s free energy. This minimisation is done with respect to the posterior over world states and with respect to action. The former corresponds to perception and ensures that the agent maintains a good estimate of the state of the world and the latter implements how the agent manipulates its environment, i.e., produces behaviour. While the former is a particular instantiation of the Bayesian brain hypothesis, and hence not necessarily a new idea, the latter had not previously been proposed and subsequently spurred some controversy (cf. above).

At this point it is important to note that the action variables are defined on the level of primitive reflex arcs, i.e., they directly control muscles in response to unexpected basic sensations. Yet, the agent can produce arbitrary complex actions by suitably setting sensory expectations which can be done via priors in the model of the agent. In comparison with reinforcement learning, the priors of the agent about states of the world (the probability mass attributed by the prior to the states), therefore, replace values or costs. But how does the agent choose its priors? This is the main question addressed by the present paper, however, only in the context of a freely exploring (i.e., task-free) agent.

In this paper, Friston et al. postulate that an agent minimises the joint entropy of world states and sensory observations instead of only the entropy of world states. Because the joint entropy is the sum of sensory entropy and conditional entropy (world states conditioned on sensory observations), the agent needs to implement two minimisations. The minimisation of sensory entropy is exactly the same as before implementing perception and action. However, conditional entropy is minimised with respect to the priors of the agent’s model, implementing higher-level action selection.

In Friston’s dynamic free energy framework (and other filters) priors correspond to predictive distributions, i.e., distributions over the world states some time in the future given their current estimate. Friston also assumes that the prior densities are Gaussian. Hence, priors are parameterised by their mean and covariance. To manipulate the probability mass attributed by the prior to the states he, thus, has to change prior mean or covariance of the world states. In the present paper the authors use a fixed covariance (as far as I can tell) and implement changes in the prior by manipulating its mean. They do this indicrectly by introducing new, independent control variables (“controls” from here on) which parameterise the dynamics of the world states without having a dynamics associated with themselves. The controls are treated like the other hidden variables in the agent model and their values are inferred from the sensory observations via free energy minimisation. However, I guess, that the idea is to more or less fix the controls to their prior means, because the second entropy minimisation, i.e., minimisation of the conditional entropy, is with respect to these prior means. Note that the controls are pretty arbitrary and can only be interpreted once a particular model is considered (as is the case for the remaining variables mentioned so far).

As with the sensory entropy, the agent has no direct access to the conditional entropy. However, it can use the posterior over world states given by the variational approximation to approximate the conditional entropy, too. In particular, Friston et al. suggest to approximate the conditional entropy using a predictive density which looks ahead in time from the current posterior and which they call counterfactual density. The entropy of this counterfactual density tells the agent how much uncertainty about the global state of the world it can expect in the future based on its current estimate of the world state. The authors do not specify how far in the future the counterfactual density looks. They here use the denotational trick to call negative conditional entropy ‘saliency’ to make the correspondence between the suggested framework and experimental variables in their example more intuitive, i.e., minimisation of conditional entropy becomes maximisation of saliency. The actual implementation of this nonlinear optimisation is computationally demanding. In particular, it will be very hard to find global optima using gradient-based approaches. In this paper Friston et al. bypass this problem by discretising the space spanned by the controls (which are the variables with respect to which they optimise), computing conditional entropy at each discrete location and simply selecting the location with minimal entropy, i.e., they do grid search.

In summary, the present paper extends previous versions of Friston’s free energy principle by adding prior selection, or, say, high-level action, to perception and action. This is done by adding new control variables representing high-level actions and setting these variables using a new optimisation which minimises future uncertainty about the state of the world. The descriptions in the paper implicitly suggest that the three processes happen sequentially: first the agent perceives to get the best estimate of the current world state, then it produces action to take the world state closer to its expectations and then it reevaluates expectations and thus sets high-level actions (goals). However, Friston’s formulations are in continuous time such that all these processes supposedly happen in parallel. For perception and action alone this leads to unexpected interactions. (Do you rather perceive the true state of the world as it is, or change it such that it corresponds to your expectations?) Adding control variables certainly doesn’t reduce this problem, if their values are inferred (perceived), too, but if perception cannot change them, only action can reduce the part of free energy contributed by them, thereby disentangling perception and action again. Therefore, the new control variables may be a necessary extension, if used properly. To me, it does not seem plausible that high-level actions are reevaluated continuously. Shouldn’t you wait until, e.g., a goal is reached? Such a mechanism is still missing in the present proposal. Instead the authors simply reevaluate high-level actions (minimise conditional entropy with respect to control variable priors) at fixed, ad-hoc intervals spanning sufficiently large amounts of time.

Review of presented experiments (saccade model)

To illustrate the theoretical points, Friston et al. present a model for saccadic eye movements. This model is very basic and is only supposed to show in principle that the new minimisation of conditional entropy can provide sensible high-level action. The model consists of two main parts: 1) the world, which defines how sensory input changes based on the true underlying state of the world and 2) the agent, which defines how the agent believes the world behaves. In this case, the state of the world is the position in a viewed image which is currently fixated by the eye of the agent. This position, hence, determines what input the visual sensors of the agent currently get (the field of view around the fixation position is restricted), but additionally there are proprioceptive sensors which give direct feedback about the position. Action changes the fixation position. The agent has a similar, but extended model of the world. In it, the fixation position depends on the hidden controls. Additionally, the model of the agent contains several images such that the agent has to infer what image it sees based on its sensory input.

In Friston’s framework, inference results heavily depend on the setting of prior uncertainties of the agent. Here, the agent is assumed to have certain proprioception, but uncertain vision such that it tends to update its beliefs of what it sees (which image) rather than trying to update its beliefs of where it looks. [I guess, this refers to the uncertainties of the hidden states and not the uncertainties of the actual sensory input which was probably chosen to be quite certain. The text does not differentiate between these and, unfortunately, the code was not yet available within the SPM toolbox at the time of writing (08.09.2012).]

As mentioned above, every 16 time steps the prior for the hidden controls of the agent is recomputed by minimising the conditional entropy of the hidden states given sensory input (minimising the uncertainty over future states given the sensory observations up to that time point). This is implemented by defining a grid of fixation positions and computing the entropy of the counterfactual density (uncertainty of future states) while setting the mean of the prior to one of the positions. In effect, this translates for the agent into: ‘Use your internal model of the world to simulate how your estimate of the world will change when you execute a particular high-level action. (What will be your beliefs about what image you see, when fixating a particular position?) Then choose the high-level action which reduces your uncertainty about the world most. (Which position gives you most information about what image you see?)’ Up to here, the theoretical ideas were self-contained and derived from first principles, but then Friston et al. introduce inhibition of return to make their results ‘more realistic’. In particular, they introduce an inhibition of return map which is a kind of fading memory of which positions were previously chosen as saccade targets and which is subtracted from the computed conditional entropy values. [The particular form of the inhibition of return computations, especially the initial substraction of the minimal conditional entropy value, is not motivated by the authors.]

For the presented experiments the authors use an agent model which contains three images as hypotheses of what the agent observes: a face and its 90° and 180° rotated versions. The first experiment is supposed to show that the agent can correctly infer which image it observes by making saccades to low conditional entropy (‘salient’) positions. The second experiment is supposed to show that, when an image is observed which is unknown to the agent, the agent cannot be certain of which of the three images it observes. The third experiment is supposed to show that the uncertainty of the agent increases when high entropy high-level actions are chosen instead of low entropy ones (when the agent chooses positions which contain very little information). I’ll discuss them in turn.

In the first experiment, the presented posterior beliefs of the agent about the identity of the observed image show that the agent indeed identifies the correct image and becomes more certain about it. Figure 5 of the paper also shows us the fixated positions and inhibition of return adapted conditional entropy maps. The presented ‘saccadic eye movements’ are misleading: the points only show the stabilised fixated positions and the lines only connect these without showing the large overshoots which occur according to the plot of ‘hidden (oculomotor) states’. Most critically, however, it appears that the agent already had identified the right image with relative certainty before any saccade was made (time until about 200ms). The results, therefore, do not clearly show that the saccade selection is beneficial for identifying the observed image, also because the presented example is only a single trial with a particular initial fixation point and with a noiseless observed image. Also, because the image was clearly identified very quickly, my guess is that the conditional entropy maps would be very similar after each saccade without inhibition of return, i.e., always the same fixation position would be chosen and no exploratory behaviour (saccades) would be seen, but this needs to be confirmed by running the experiment without inhibition of return. My overall impression of this experiment is that it presents a single, trivial example which does not allow me to draw general conclusions about the suggested theoretical framework.

The second experiment acts like a sanity check: the agent shouldn’t be able to identify one of its three images, when it observes a fourth one. Whether the experiment shows that, depends on the interpretation of the inferred hidden states. The way these states were defined their values can be directly interpreted as the probability of observing one of the three images. If only these are considered the agent appears to be very certain at times (it doesn’t help that the scale of the posterior belief plot in Figure 6 is 4 times larger than that of the same plot in Figure 5). However, the posterior uncertainty directly associated with the hidden states appears to be indeed considerably larger than in experiment 1, but, again, this is only a single example. Something that is rather strange: the sequence of fixation positions is almost exactly the same as in experiment 1 even though the observed image and the resulting posterior beliefs were completely different. Why?

Finally, experiment three is more like a thought experiment: what would happen, if an agent chooses high-level actions which maximise future uncertainty instead of minimising it. Well, the uncertainty of the agent’s posterior beliefs increases as shown in Figure 7, which is the expected behaviour. One thing that I wonder, though, and it applies to the presented results of all experiments: In Friston’s Bayesian filtering framework the uncertainty of the posterior hidden states is a direct function of their mean values. Hence, as long as the mean values do not change, the posterior uncertainty should stay constant, too. However, we see in Figure 7 that the posterior uncertainty increases even though the posterior means stay more or less constant. So there must be an additional (unexplained) mechanism at work, or we are not shown the distribution of posterior hidden states, but something slightly different. In both cases, it would be important to know what exactly resulted in the presented plots to be able to interpret the experiments in the correct way.

Conclusion

The paper presents an important theoretical extension to Friston’s free energy framework. This extension consists of adding a new layer of computations which can be interpreted as a mechanism for how an agent (autonomously) chooses its high-level actions. These high-level actions are defined in terms of desired future states encoded by the probability mass which is assigned to these states by the prior state distribution. Conceptually, these ideas translate into choosing maximally informative actions given the agent’s model of the world and its current state estimate. As discussed by Friston et al. such approaches to action selection are not new (see also Tishby and Polani, 2011). So, the author’s contribution is to show that these ideas are compatible with Friston’s free energy framework. Hence, on the abstract, theoretical level this paper makes sense. It also provides a sound theoretical argument for why an agent would not seek sensory deprivation in a dark room, as feared by critics of the free energy principle. However, the presented framework heavily relies on the agent’s model of the world and it leaves open how the agent has attained this model. Although the free energy principle also provides a way for the agent to learn parameters of its model, I still, for example, haven’t seen a convincing application in which the agent actually learnt the dynamics of an unknown process in the world. Probably Friston would here also refer to evolution as providing a good initialisation for process dynamics, but I find that a too cheap way out.

From a technical point of view the paper leaves a few questions open, for example: How far does the counterfactual distribution look into the future? What does it mean for high-level actions to change how far the agent looks into his subjective future? How well does the presented approach scale? Is it important to choose the global minimum of the conditional entropy (this would be bad, as it’s probably extremely hard to find in a general setting)? When, or how often, does the agent minimise conditional entropy to set high-level actions? What happens with more than one control variables (several possible high-level actions)? How can you model discrete high-level actions in Friston’s continuous Gaussian framework? How do results depend on the setting of prior covariances / uncertainties. And many more.

Finally, I have to say that I find the presented experiments quite poor. Although providing the agent with a limited field of view such that it has to explore different regions of a presented image is a suitable setting to test the proposed ideas, the trivial example and introduction of ad-hoc inhibition of return make it impossible to judge whether the underlying principle is successfully at work, or the simulations have been engineered to work in this particular case.

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.

Sum-Product Networks: A New Deep Architecture.

Poon, H. and Domingos, P.
in: Proceedings of the 27th conference on Uncertainty in Artificial Intelligence (UAI 2011), 2011
URL, Google Scholar

Abstract

The key limiting factor in graphical model inference and learning is the complexity of the partition function. We thus ask the question: what are general conditions under which the partition function is tractable? The answer leads to a new kind of deep architecture, which we call sumproduct networks (SPNs). SPNs are directed acyclic graphs with variables as leaves, sums and products as internal nodes, and weighted edges. We show that if an SPN is complete and consistent it represents the partition function and all marginals of some graphical model, and give semantics to its nodes. Essentially all tractable graphical models can be cast as SPNs, but SPNs are also strictly more general. We then propose learning algorithms for SPNs, based on backpropagation and EM. Experiments show that inference and learning with SPNs can be both faster and more accurate than with standard deep networks. For example, SPNs perform image completion better than state-of-the-art deep networks for this task. SPNs also have intriguing potential connections to the architecture of the cortex.

Review

The authors present a new type of graphical model which is hierarchical (rooted directed acyclic graph) and has a sum-product structure, i.e., the levels in the hierarchy alternately implement a sum or product operation of their children. They call these models sum-product networks (SPNs). The authors define conditions under which SPNs represent joint probability distributions over the leaves in the graph efficiently where efficient means that all the marginals can be computed efficiently, i.e., inference in SPNs is easy. They argue that SPNs subsume all previously known tractable graphical models while being more general.

When inference is tractable in SPNs, so is learning. Learning here means to update weights in the SPN which can also be used to change the structure of an SPN by pruning connections with 0 weights after convergence of learning. They suggest to use either EM or gradient-based learning, but note that for large hierarchies (very deep networks) you’ll have a gradient diffusion problem, as in general in deep learning. To overcome this problem they use the maximum posterior estimator which effectively updates only a single edge of a node instead of all edges dependent on the (diffusing) gradient.

The authors introduce the properties of SPNs using only binary variables. Leaves of the SPNs then are indicators for values of these variables, i.e., there are 2*number of variables leaves. It is straight forward to extend this to general discrete variables where the potential number of leaves then rises to number of values * number of variables. For continuous variables sum nodes become integral nodes (so you need distributions which you can easily integrate) and it is not so clear to me what leaves in the tree then are. In general, I didn’t follow the technical details well and can hardly comment on potential problems. One question certainly is how you initialise your SPN structure before learning (it will matter whether you start with a product or sum level at the bottom of your hierarchy and where the leaves are positioned).

Anyway, this work introduces a promising new deep network architecture which combines a solid probabilistic interpretation with tractable exact computations. In particular, in comparison to previous models (deep belief networks and deep Boltzmann machines) this leads to a jump in performance in both computation time and inference results as shown in image completion experiments. I’m looking forward to seeing more about this.

Flexible vowel recognition by the generation of dynamic coherence in oscillator neural networks: speaker-independent vowel recognition.

Liu, F., Yamaguchi, Y., and Shimizu, H.
Biol Cybern, 71:105–114, 1994
DOI, Google Scholar

Abstract

We propose a new model for speaker-independent vowel recognition which uses the flexibility of the dynamic linking that results from the synchronization of oscillating neural units. The system consists of an input layer and three neural layers, which are referred to as the A-, B- and C-centers. The input signals are a time series of linear prediction (LPC) spectrum envelopes of auditory signals. At each time-window within the series, the A-center receives input signals and extracts local peaks of the spectrum envelope, i.e., formants, and encodes them into local groups of independent oscillations. Speaker-independent vowel characteristics are embedded as a connection matrix in the B-center according to statistical data of Japanese vowels. The associative interaction in the B-center and reciprocal interaction between the A- and B-centers selectively activate a vowel as a global synchronized pattern over two centers. The C-center evaluates the synchronized activities among the three formant regions to give the selective output of the category among the five Japanese vowels. Thus, a flexible ability of dynamical linking among features is achieved over the three centers. The capability in the present system was investigated for speaker-independent recognition of Japanese vowels. The system demonstrated a remarkable ability for the recognition of vowels very similar to that of human listeners, including misleading vowels. In addition, it showed stable recognition for unsteady input signals and robustness against background noise. The optimum condition of the frequency of oscillation is discussed in comparison with stimulus-dependent synchronizations observed in neurophysiological experiments of the cortex.

Review

The authors present an oscillating recurrent neural network model for the recognition of Japanese vowels. The model consists of 4 layers: 1) an input layer which gives pre-processed frequency information, 2) an oscillatory hidden layer with local inhibition, 3) another oscillatory hidden layer with long-range inhibition and 4) a readout layer implementing the classification of vowels using a winner-takes-all mechanism. Layers 1-3 each contain 32 units where each unit is associated to one input frequency. The output layer contains one unit for each of the 5 vowels and the readout mechanism is based on multiplication of weighted sums of layer 3 activities such that the output is also oscillatory. The oscillatory units in layers 2 and 3 consist of an excitatory element coupled with an inhibitory element which oscillate, or become silent, depending on the input. The long-range connections in layer 3 are determined manually based on known correlations between formants (characteristic frequencies) of the individual vowels.

In experiments the authors show that the classification of their network is robust against different speakers (14 men, 5 women, 5 girls, 5 boys): 6 out of 145 trials were correctly classified. However, they do not report what exactly their criterion for classification performance was (remember that the output was oscillatory, also sometimes alternative vowels show bumps in the time course of a vowel in the shown examples). They also report robustness to imperfect stimuli (formants varying within a vowel) and noise (superposition of 12 different conversations), but only single examples are shown.

Without being able to tell what the state of the art in neural networks in 1994 was, I guess the main contribution of the paper is that it shows that vowel recognition may be robustly implemented using oscillatory networks. At least from today’s perspective the suggested network is a bad solution to the technical problem of vowel recogntion, but even alternative algorithms at the time were probably better in that (there’s a hint in one of the paragraphs in the discussion). The paper is a good example for what was wrong with neural network research at the time: the models give the feeling that they are pretty arbitrary. Are the units in the network only defined and connected like they are, because these were the parameters that worked? Most probably. At least here the connectivity is partly determined through some knowledge of how frequencies produced by vowels relate, but many other parameters appear to be chosen arbitrarily. Respect to the person who made it work. However, the results section is rather weak. They only tested one example of a spoken vowel per person and they don’t define classification performance clearly. I guess, you could argue that it is a proof-of-concept of a possible biological implementation, but then again it is still unclear how this can be properly related to real networks in the brain.

Recurrent neuronal circuits in the neocortex.

Douglas, R. J. and Martin, K. A. C.
Curr Biol, 17:R496–R500, 2007
DOI, Google Scholar

Abstract

In this Primer, we shall describe one interesting property of neocortical circuits – recurrent connectivity – and suggest what its computational significance might be.

Review

First, they use data of the distribution of synapses in cat visual cortex to argue that the predominant drive of activity in a cortical area is from recurrent connections within this area. They then suggest that the reason for this is the ability to enhance and denoise incoming signals through suitable recurrent connections. They show corresponding functional behaviour in a model based on linear threshold neurons (LTNs). They do not use sigmoid activation functions, because neurons apparently only rarely operate on their maximum firing rate such that sigmoid activation functions are not necessary. To maintain stability they instead use a global inhibitory unit. I guess you could equivalently use a suitable sigmoid function. Finally they suggest that top-down connections may bias the activity in the recurrent network such that one of a few alternative inputs may be selected based on, e.g., attention.

So here the functional role of the recurrent neural network is merely to increase the signal to noise ratio. It’s a bit strange to me that actually no computation is done. Does that mean that all the computation from sensory signals to hidden states are done by the projections from lower level area to higher level area? This seems to be consistent with the reservoir computing idea where the reservoir can also be seen as enhancing the representation of the input (by stretching its effects in time). The difference just being that the dynamics and function in reservoirs is more involved.

The ideas presented here are almost the same as already proposed by the first author in 1995 (see Douglas1995).

Temporal sparseness of the premotor drive is important for rapid learning in a neural network model of birdsong.

Fiete, I. R., Hahnloser, R. H. R., Fee, M. S., and Seung, H. S.
J Neurophysiol, 92:2274–2282, 2004
DOI, Google Scholar

Abstract

Sparse neural codes have been widely observed in cortical sensory and motor areas. A striking example of sparse temporal coding is in the song-related premotor area high vocal center (HVC) of songbirds: The motor neurons innervating avian vocal muscles are driven by premotor nucleus robustus archistriatalis (RA), which is in turn driven by nucleus HVC. Recent experiments reveal that RA-projecting HVC neurons fire just one burst per song motif. However, the function of this remarkable temporal sparseness has remained unclear. Because birdsong is a clear example of a learned complex motor behavior, we explore in a neural network model with the help of numerical and analytical techniques the possible role of sparse premotor neural codes in song-related motor learning. In numerical simulations with nonlinear neurons, as HVC activity is made progressively less sparse, the minimum learning time increases significantly. Heuristically, this slowdown arises from increasing interference in the weight updates for different synapses. If activity in HVC is sparse, synaptic interference is reduced, and is minimized if each synapse from HVC to RA is used only once in the motif, which is the situation observed experimentally. Our numerical results are corroborated by a theoretical analysis of learning in linear networks, for which we derive a relationship between sparse activity, synaptic interference, and learning time. If songbirds acquire their songs under significant pressure to learn quickly, this study predicts that HVC activity, currently measured only in adults, should also be sparse during the sensorimotor phase in the juvenile bird. We discuss the relevance of these results, linking sparse codes and learning speed, to other multilayered sensory and motor systems.

Review

They model the generation of bird song as a simple feed-forward network and show that a sparse temporal code of HVC neurons (feeding into RA neurons) speeds up learning with backpropagation. They argue that this speed up is the main explanation for why real HVC neurons exhibit a sparse temporal code.

HVC neurons are modelled as either on or off, i.e., bursting or non-bursting, while RA neurons have continuous activities. A linear combination of RA neurons then determines the output of the network. They define a desired, low-pass filtered output that should be learnt, but while their Fig. 2 suggests that they model the sequential aspect of the data, the actual network has no such component and the temporal order of the data points is irrelevant for learning. Maybe fixing, i.e., not learning, the connections from RA to output is biologically well motivated, but other choices for the network seem to be quite arbitrary, e.g., why do RA neurons project from the beginning to only one of two outputs? They varied quite a few parameters and found that their main result (learning is faster with sparse HVC firing) holds for all of them, though. Interesting to note: they had to initialise HVC-RA and RA thresholds such that initial RA activity is low and nonuniform in order to get desired type of RA activity after learning.

I didn’t like the paper that much, because they showed the benefit of sparse coding for the biologically implausible backpropagation learning. Would it also hold up against a Hebbian learning paradigm? On the other hand, the whole idea of being able to learn better when each neuron is only responsible for one restricted part of the stimulus is so outrageously intuitive that you wonder why this needed to be shown in the first place (Stefan noted, though, that he doesn’t know of work investigating temporal sparseness compared to spatial sparseness)? Finally, you cannot argue that this is the main reason why HVC neurons fire in a temporally sparse manner, because there might be other unknown reasons and this is only a side effect.