Linear dimensionality reduction techniques are widely used in neuroscience to study how behaviourally-relevant variables are represented in the neurons. The general approach goes like this – (i) apply dimensionality reduction e.g. PCA on trial-averaged activity of a population of neurons to identify a -dimensional subspace () capturing a sufficient fraction of neural activity, and (ii) examine how neural dynamics evolve within this subspace to (hopefully) gain insights about neural computation. This recipe has largely been successful (ignoring failures that generally go unpublished): the reduced dimensionality of neural datasets is often quite small and the corresponding low-dimensional dynamical portraits are usually interpretable. However, neuroscientists observe only a tiny fraction of the complete neural population. So could the success of dimensionality reduction be an artefact of severe subsampling? This is precisely the question that Gao et al. attempt to answer in their paper.

They first develop a theory that describes how neural dimensionality (defined below) is bounded by the task design and some easy-to-measure properties of neurons. Then they adapt the mathematical theory of random projection to neuroscience setting and obtain the amount of geometric distortion in the neural trajectories introduced by subsampling, or equivalently, the minimum number of neurons one has to measure in order to achieve an arbitrarily small distortion in a real experiment. Throughout this post, I use the term *neural dimensionality* in the same sense that the authors use in the paper: the dimension of the smallest affine subspace that contains a large (~80 – 90%) fraction of the neural trajectories. Note that this notion of dimensionality differs from the intrinsic dimensionality of the neural manifold, which is usually much smaller.

To derive an analytical expression for dimensionality, the authors note that there is an inherent biological limit to how fast the neural trajectory can evolve as a function of the task parameters. Concretely, consider the response of a population of visual neurons to an oriented bar. As you change the *orientation* from 0 to , the activity of the neural population will likely change too. If denotes the minimum change in orientation required to induce an appreciable change in the population activity (*i.e.* the width of the autocorrelation in the population activity pattern), then the population will be able to explore roughly linear dimensions. Of course, the scale of autocorrelation will differ across brain areas (presumably increases as one goes from the retina to higher visual areas), so the neural dimensionality would depend on the properties of the population being sampled, not just on the task design. Similar reasoning applies to other task parameters such as *time* (yes, they consider *time* as a task parameter because, after all, neural activity is variable in time). If you wait for time period , the dimensionality will be roughly equal to where is now the width of temporal autocorrelation. For the general case of different task parameters, they prove that neural dimensionality is ultimately bounded by (even if you record from millions of neurons):

where is the range of the task parameter, is the corresponding autocorrelation length and is an constant which they prove is close to 1. The numerator and denominator depend on task design and smoothness of neural dynamics respectively, so they label the term on the right-hand side *neural task complexity* (NTC). This terminology was a source of confusion among some of us as it appears to downplay the fundamental role of the neural circuit properties in restricting the dimensionality, but its intended meaning is pretty clear if you read the paper.

To derive NTC, the authors assume that the neural response is *stationary* in the task parameters and the joint autocorrelation function is *factorisable* as a product of individual task parameters’ autocorrelation functions, and then show that the above bound becomes weak when these assumptions do not hold for the particular population being studied. The proof was also facilitated in part by a clever choice of the definition of dimensionality: ‘participation ratio’ where are the eigenvalues of the neuronal covariance matrix, instead of the more common but analytically cumbersome measure based on ‘fraction of variance explained’ , but they demonstrate that their choice is reasonable.

Much of the discussion in our journal club centred on whether equation (1) is just circular reasoning, and whether we really gain any new insight from this theory. This view was somewhat understandable because the authors introduce the paper by promising to present a theory that explains the origin of the simplicity betrayed by the low dimensionality of neural recordings… only to show us that it emerges from the specific way in which neural populations respond (smooth dynamics large denominator) to specific tasks (low complexity small numerator). Although this result may seem qualitatively trivial, the strength of their work lies in making our intuitions precise and packaging them in the form of a compact theorem. Moreover, as shown later in the paper, knowing this bound on dimensionality can be practically helpful in determining how many neurons to record. Before discussing that aspect, I’d like to briefly dwell a little bit on a potentially interesting corollary and a possible extension of the above theorem.

Based on the above theorem, one can identify three regimes of dimensionality for a recording size of neurons:

(i)

(ii)

(iii)

The first two regimes are pretty straightforward to interpret. (i) implies that you might not have sampled enough neurons, while (ii) means that the task was not complex enough to elicit richer dynamics. The authors call (iii) the most interesting and say ‘*Then, and only then, can one say that the dimensionality of neural state space dynamics is constrained by neural circuit properties above and beyond the constraints imposed by the task and smoothness of dynamics alone’*. What could those properties be? Here, it is worth noting that their theory takes the *speed* of neural dynamics into account, but not the *direction*. Recurrent connections, for example, might prevent the neural trajectory from wandering in certain directions thereby constraining the dimensionality. Such constraints may in fact lead to nonstationary and/or unfactorisable neuronal covariance, violating the conditions that are necessary for dimensionality to approach NTC. Although this is not explicitly discussed, they simulate a non-normal network to demonstrate that its dimensionality is reduced by recurrent amplification. So I guess it must be possible to derive a stronger theorem with a tighter bound on neural dimensionality by incorporating the influence of the strength and structure of connections between neurons.

NTC is a bound on the size of the linear subspace within which neural activity is mostly confined. But even if NTC is small, it is not clear whether we can accurately estimate the neural trajectory within this subspace simply by recording neurons such that . After all, is still only a tiny fraction of the total number of neurons in the population . To explore this, the authors use the theory of random projection and show that it is possible to achieve some desired level of fractional error in estimating the neural trajectory by ensuring:

where is the number of task parameters. This means that the demands on the size of the neural recording grow only linearly in the number of task parameters and logarithmically (!!) in both NTC and . Equation (2) holds as long as the recorded sample is statistically homogenous to the rest of the neurons, a restriction that is guaranteed for most higher brain areas provided the sampling is unbiased i.e. the experimenter does not cherry-pick which neurons to record/analyse. The authors encourage us to use their theorems to obtain back-of-the-envelope estimates of recording size and to guide experimental design. This is easier said than done, especially when studying a new brain area or when designing a completely new task. Nevertheless, their work is likely to push the status quo in neuroscience experiments by encouraging experimentalists to move boldly towards more complex tasks without radically revising their approach to neural recordings.

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probe whether rodents can optimally discount evidence by adapting the timescale over which they accumulate it.

Sensory

excess-rate

(really to jump in evidence at each click)

In conclusion, we believe that the *dynamic clicks task* experiment described in this paper is key for the study of adaptive evidence accumulation. The reported evidence that some rats are able to change their evidence discounting rate according to the environment’s volatility is convincing. On the theoretical side, we wonder whether additional models could produce similar reverse correlation curves, and this could represent a route for further research projects.

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Predictive coding is used to describe very different approaches in the neuroscience literature [3]. This paper understood the term as representing prediction error by neural responses. Alternatively, it has been defined as neurons that preferentially encode stimuli that carry information about the future [4]. The two definitions are not necessarily consistent with each other, since the neurons representing errors might not be predictive about the future.

Bayesian inference uses the sensory input to compute the posterior over latent variables. Based on this posterior, the brain might pick an ideal point estimate about the latent. This preferred latent could then be used to compute a prediction about input.

Predictive coding doesn’t specify how to generate predictions, whereas Bayesian inference offers a natural way to compute the prediction. It seems natural to combine them to create Bayesian predictive coding. In [5], a hierarchical model that use neurons to carry out the prediction error generated by the bottom-up inputs and top-down predictions is built to model the vision system. Several experimental results (e.g. surrounding suppression) can be explained by this hierarchical Bayesian predictive coding model. But this model also used neurons to represent the prediction and latent variables. Thus it’s a more like a hybrid Bayesian predictive and direct variable coding model. We then argued in our journal club about if the pure predictive coding idea is enough to represent the brain’s inference process. Mathematically, this can be understood as follows. If the prior and likelihood are all Gaussians, maximization of posterior with respect to weights is equivalent to the minimization of the sum-of-square error function. If the brain uses gradient descent to perform learning, the gradient of sum-of-square error function will be dependent on the prediction errors. This means the brain need to represent the prediction errors if it is doing optimal Bayesian inference under Gaussian assumption. However, the brain might do suboptimal Bayesian inference. The distribution of input can also be non-Gaussian. Thus in general, only representing prediction error is not enough for the brain to perform Bayesian inference.

One advantage of predictive coding is the brain can save neural representational power by just represent the prediction error. However, this idea doesn’t consider the representation of prediction itself. The brain still must spend neural resources to represent the prediction. The efficient coding framework [6] claims that neural representations are trying to maximize the mutual information between responses and stimulus under some constraints. Under high signal-to-noise ratios, the efficient representation will meet with the representation learned from predictive coding principle [7,8]. However, when the signal to noise ratio is small, the efficient coding says that preserving all the stimulus input is more beneficial compared to just preserving the prediction error [7,8]. Thus this predictive coding motif isn’t always true under the efficient coding framework.

In summary, we agreed with the conclusions of this paper. More experiments need to be designed to offer conclusive answers about how the brain performs Bayesian inference. We also can’t rule out the possibility that brain use multiple representations. Unifying such possible representations in a more general theoretical framework would be worth the future efforts of computational neuroscientists.

[1]. Aitchison L, Lengyel M. With or without you: predictive coding and Bayesian inference in the brain. Current Opinion in Neurobiology, 2017, 46: 219-227.

[2]. Marr D: Vision: a computational investigation into the human representation and processing of visual information. WH San Francisco: Freeman and Company; 1982.

[3]. Chalk M, Marre O, Tkacik G. Towards a unified theory of efficient, predictive and sparse coding. bioRxiv, 2017: 152660.

[4]. Bialek W, Van Steveninck R R D R, Tishby N. Efficient representation as a design principle for neural coding and computation[C]//Information Theory, 2006 IEEE International Symposium on. IEEE, 2006: 659-663.

[5]. Rao RP, Ballard DH: Predictive coding in the visual cortex: a functional interpretation of some extra-classical receptive-field effects. Nat Neurosci 1999, 2:79-87.

[6]. Barlow H: Possible principles underlying the transformations of sensory messages. In Sensory Communication. Edited by Rosenblith W. MIT Press; 1961:217-234.

[7]. Atick J J, Redlich A N. Towards a theory of early visual processing. Neural Computation, 1990, 2(3): 308-320.

[8]. Srinivasan M V, Laughlin S B, Dubs A. Predictive coding: a fresh view of inhibition in the retina. Proceedings of the Royal Society of London B: Biological Sciences, 1982, 216(1205): 427-459.

]]>Inferring connectivity from activity is not a new idea: Cross-correlation functions measure the average impact of one cells’ spike on the activity of another directly from their *concurrently* measured spike trains. The idea that cross-correlations can be used to infer synaptic interactions between cells goes back to at least 1970 [4]. However, this early work also recognized several difficulties of this approach. Cross-correlations will reflect common inputs to the cells, and global patterns of activity of the observed populations. Disentangling synaptic interactions from these other effects is difficult, especially if only a fraction of a population is observed.

Assuming that the activity of an entire population has been observed, one approach to disentangling direct from indirect interactions between cells is to use partial correlations: the correlations between the residuals of two cells’ activities remaining after regression on the other cells’ activities, . In other words, partial correlations are the correlations that remain between two neurons when their correlations with all other cells are removed. Again, this is an old idea [5], and other approaches have been proposed to tackle the problem: the connectivity inferred by fitting Ising models, generalized linear models, and other types of models have been proposed to uncover synaptic interactions.

In our recent journal club we discussed a recent addition by Kadirvelu, et al [6] to this fairly extensive body of literature. Here the authors asked how well thresholded partial correlations and thresholded weights obtained from fitting an Ising model can represent synaptic connectivity. The authors first simulated networks of 11 to 120 Izhikevich neurons under varying conditions, changing the firing rates, connectivity structure, etc., of the network. They then tried to recover the connectivity using the two methods, and compare the results to the actual ground truth used in the simulations. Synaptic weights were deemed unimportant, and instead binary matrices with 0s and 1s signifying the absence or presence of an interaction, respectively, were compared. As partial correlations do not reveal the direction of an interaction, the ground truth matrices were symmetrized before a comparison. The performance of each method was quantified by the area under the ROC curve obtained from varying the threshold. Low thresholds gives more false positives, and high thresholds more false negatives. Thus as the threshold is changed from low to high, both the fraction of falsely identified synaptic connections (false positives, FP), and the fraction of correctly identified connections (true positives, TP) both increase. The curve traced out by the false and true positive rate in FP-TP space is the ROC curve.

The main conclusion of the paper is that the performance of the methods depends on the level of correlations: At low correlations, fitting an Ising model works better, and at high correlations the partial correlation method works better. Other observations were not unexpected: increasing the firing rates improves inference (as the number of “interactions”, i.e. spikes increases). As the number of neurons increases, inference was harder, etc.

One has to ask what testing these models using simulations can tell us: These settings are highly idealized, and miss many of the features one would encounter with real data. One of the main issues is that latent inputs are not accounted for. In this particular case, all correlations were due to synaptic interactions between model cells, and all cells were observed. Global fluctuations can also induce strong correlations [7], completely overshadowing the effects of direct interactions [1]. There are many other subtleties: the direct inversion of the correlation matrix to obtain partial correlations is problematic, and typically some regularization is required [8]. Moreover, thresholding of inferred interaction weights to try to distinguish real interactions from fluctuations is known to give inconsistent estimators of interactions.

So is the inference of interactions a futile exercise? With the present data, inferring synaptic interactions is likely to be unsuccessful in all but the simplest settings. However, robustly inferring the strength of interactions is still worthwhile, even if these only measure statistical dependencies, rather than structural connections. Changes in such effective connectivity may reflect computations or mental states, and are hypothesized to change under working memory load [9]. Moreover, the effective connectivity could be modulated much more quickly than synaptic connectivity. However, as to which method is best at robustly uncovering such effective connectivity, the article we discussed is silent.

1. Rosenbaum, R., Smith, M. A., Kohn, A., Rubin, J. E., & Doiron, B. (2016). The spatial structure of correlated neuronal variability. Nature Neuroscience, 20(1), 107–114.

2. Jiang, X., Shen, S., Cadwell, C. R., Berens, P., Sinz, F., Ecker, A. S., et al. (2015). Principles of connectivity among morphologically defined cell types in adult neocortex. Science 350(6264).

3. Oswald, A.-M. M., & Reyes, A. D. (2008). Maturation of intrinsic and synaptic properties of layer 2/3 pyramidal neurons in mouse auditory cortex. Journal of Neurophysiology, 99(6), 2998–3008.

4. Moore, G. P., Segundo, J. P., Perkel, D. H., & Levitan, H. (1970). Statistical signs of synaptic interaction in neurons. Biophysical Journal, 10(9), 876–900.

5. Brillinger, D. R., Bryant, H. L., & Segundo, J. P. (1976). Identification of synaptic interactions. Biological Cybernetics, 22(4), 213–228.

6. Kadirvelu, B., Hayashi, Y., & Nasuto, S. J. (2017). Inferring structural connectivity using Ising couplings in models of neuronal networks. Scientific Reports, 7(1), 8156.

7. Ecker, A. S., Denfield, G. H., Bethge, M., & Tolias, A. S. (2015). On the structure of population activity under fluctuations in attentional state.

8. Yatsenko, D., Josić, K., Ecker, A. S., Froudarakis, E., Cotton, R. J., & Tolias, A. S. (2015). Improved estimation and interpretation of correlations in neural circuits. PLoS Computational Biology, 11(3), e1004083.

9. Pinotsis, D. A., Buschman, T. J., & Miller, E. K. (n.d.). Working Memory Load Modulates Neuronal Coupling.

]]>As the title suggests, the authors aim to synthesize bayesian inference with efficient coding. The Bayesian brain hypothesis states that the brain computes posterior probabilities based on its model of the world (prior) and its sensory measurements (likelihood). Efficient coding assumes that the brain distributes its resources to maximize a cost, typically information. In particular, they note that efficient coding that optimizes mutual information is a special case of their more general framework, but ask whether other maximizations based on the Bayesian posterior might better explain data.

Denoting stimulus , measurements , and model parameters , they use the following ingredients for their theory: a prior , a likelihood , an encoding capacity constraint , and a loss functional . They assume that the brain is able to construct the true posterior . The goal is to find a model that optimizes the expected loss

under the constraint .

The loss functional is the key. The authors consider two things the loss might depend on: the posterior , or the ground truth . They needed to make the loss explicitly dependent on the posterior in order to optimize for mutual information. It was unclear whether they also considered a loss depending on *both*, which seems critical. We communicated with them and they said they’d clarify this in the next version.

They state that there is no clear *a priori* reason to maximize mutual information (or equivalently to minimize the average posterior entropy, since the prior is fixed). They give a nice example of a multiple choice test for which encodings that maximize information will achieve fewer correct answers than encodings that maximize percent correct for the MAP estimates. The ‘best’ answer depends on how one defines ‘best’.

After another few interesting gaussian examples, they revisit the famous Laughlin (1981) result on efficient coding in the blowfly. This was hailed as a triumph for efficient coding theory in predicting the nonlinear input-output photoreceptor curve derived directly from the measured prior over luminance. But here the authors found that instead a different loss function on the posterior gave a better fit. Interestingly, though, that loss function was based on a point estimate,

where the point estimate is the Bayesian optimum for this cost function and is a parameter. The limit gives the familiar entropy, is the conventional squared error, and the best fit to the data was , a “square root loss.” It’s hard to provide any normative explanation of *why* this or any other choice is best (since the loss is basically the definition of ‘best’, and you’d have to relate the theoretical loss to some real consequences in the world), it is very interesting that the efficient coding solution explains data worse than their other Bayesian efficient coding losses.

Besides the minor confusion about whether their loss does/should include the ground truth , and some minor disagreement about how much others have done things along this line (Ganguli and Simoncelli, Wei and Stocker, whom they do cite), my biggest question is whether the cost really should depend on the posterior as opposed to a point estimate. I’m a fan of Bayesianism, but ultimately one must take a single action, not a distribution. I discussed this with Jonathan over email, and he maintained that it’s important to distinguish an action from a point estimate of the stimulus: there’s a difference between the width of the river and whether to try to jump over it. I countered that one could refer actions back to the stimulus: the river is jumpable, or unjumpable (essentially a Gibsonian affordance). In a world of latent variables, any point estimate based on a posterior is a compromise based on the loss function.

So when should you keep around a posterior, rather than a point estimate? It may be that the appropriate loss function changes with context, and so the best point estimate would change too. While one could certainly consider that to be a bigger computation to produce a context-dependent point estimate, it may be more parsimonious to just represent information about the posterior directly.

]]>Contact: KiJung [dot] Yoon [at] bcm [dot] edu

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