Stable task information from an unstable neural population

In this paper, we test one of the hypothesis from the "Causes and consequences of representational drift" review: that neural population codes might support a stable readout, even as single neural tunings reconfigure. 

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We examined long-term neuroimaging recordings form Driscoll et al. (2017). We found that the population code does, indeed, preserve a stable readout over time. This is because much of the reconfiguration in the neural code is orthogonal to the directions that encode behavior. 

Even this readout isn't perfectly stable, however, so some amount of ongoing compensation must occur to allow readouts to adjust to the shifting neural code. This residual reconfiguration is much slower than the apparent day-to-day variability in single neurons, so it could be easily tracked with synaptic plasticity. 

Many thanks to all! This really was a team effort: data were collected by Laura Driscoll and colleagues in the Harvey lab, and most of the analysis was started by Adriana Loback. Dhruva Raman was instrumental in sorting out the null model simulations. The paper can be cited as

Rule, M.E., Loback, A.R., Raman, D.V., Driscoll, L.N., Harvey, C.D. and O'Leary, T., 2020. Stable task information from an unstable neural population. Elife, 9, p.e51121.

Excerpt: Figure 4

A slowly-varying component of drift disrupts the behavior-coding subspace. (a) The small error increase when training concatenated decoders (Figure 3) suggests that plasticity is needed to maintain good decoding in the long term. We assess the minimum rate for this plasticity by training a separate decoder Md for each day, while minimizing the change in weights across days. The parameter λ controls how strongly we constrain weight changes across days (the inset equation reflects the objective function to be minimized; Methods). (b) Decoders trained on all days (cyan) perform better than chance (red), but worse than single-day decoders (ochre). Black traces illustrate the plasticity-accuracy trade-off for adaptive decoding. Modest weight changes per day are sufficient to match the performance of single-day decoders (Boxes: inner 50% of data, horizontal lines: median, whiskers: 5–95th%). (c) Across days, the mean neural activity associated with a particular phase of the task changes (Δμ). We define an alignment measure ρ (Materials and methods) to assess the extent to which these changes align with behavior-coding directions in the population code (blue) verses directions of noise correlations (ochre). (d) Drift is more aligned with noise (ochre) than it is with behavior-coding directions (blue). Nevertheless, drift overlaps this behavior-coding subspace much more than chance (grey; dashed line: 95% Monte-Carlo sample). Each box reflects the distribution over all maze locations, with all consecutive pairs of sessions combined.

Causes and consequences of representational drift

Drs. Harvey, O'Leary, and myself, have just published a review in Current Opinion in Neurobiology titled the "Causes and consequences of representational drift" [PDF].

We explore recent results from the work of Alon Rubin and Liron Sheintuch in the Ziv lab and Laura Driscoll in the Harvey lab. Their work has shown that neural representations reconfigure even for fixed, learned tasks. 

We discuss ways that the brain might support this reconfiguration without forgetting what it has already learned. There might be a subset of stable neurons that maintain memories. Alternatively, neural representations can be highly redundant, and support a stable readout even as single cells seem to change. 

Finally, we conjecture that redundancy allows different brain areas to "error-correct" each other, allowing the brain to keep track of the shifting meaning of single neurons in plastic representations.

For a brief preview, here are the pre-publication versions of figures 2 and 3:

Figure 2:

Figure 2: Internal representations have unconstrained degrees of freedom that allow drift. (a) Nonlinear dimensionality reduction of population activity recovers the low-dimensional structure of the T-maze in Driscoll et al. (2017). Each point represents a single time-point of population activity, and is colored according to location in the maze. (b) Point clouds illustrate low-dimensional projections of neural activity as in (a). Although unsupervised dimensionality-reduction methods can recover the task structure on each day, the way in which this structure is encoded in the population can change over days to weeks. (c) Left: Neural populations can encode information in relative firing rates and correlations, illustrated here as a sensory variable encoded in the sum of two neural signals ($y_1 + y_2$). Points represent neural activity during a repeated presentation of the same stimulus. Variability orthogonal to this coding axis does not disrupt coding, but could appear as drift in experiments if it occurred on slow timescales. Right: Such distributed codes may be hard to read-out from recorded subpopulations (e.g. $y_1$ or $y_2$ alone; black), especially if they entail correlations between brain areas. (d) Left: External covariates may exhibit context-dependent relationships. Each point here reflects a neural population state at a given time-point. The relationship between directions $x_1$ and $x_2$ changes depending on context (cyan versus red). Middle: Internally, this can be represented a mixture model, in which different subspaces are allocated to encode each context, and the representations are linearly-separable (gray plane). Right: The expanded representation contains two orthogonal subspaces that each encode a separate, context- dependent relationship. This dimensionality expansions increases the degrees of freedom in internal representations, thereby increasing opportunities for drift.

 

Constrained plasticity can compensate for ongoing drift in neural populations

I've started a new postdoc, working on a collaboration between the O'Leary, Ziv, and Harvey labs, on the Human Frontiers Science Program grant, "Building a theory of shifting representations in the mammalian brain". 

To start, I've been working with Adrianna Loback to try to make sense of a puzzling result from Driscoll et al. (2017): the neural code for sensorimotor variables in parietal cortex is unstable, changing dramatically even for habitual tasks in which no learning takes place.

We think the brain might be using a distributed population code. Because there are so many possible ways to read-out a distributed and redundant population code, it could be that there is a stable representation at the population level, despite the apparent instability of single neurons. 

I'll be presenting our work to date as a poster at the UK Neural Computation conference in Nottingham, July 1st-3rd. 

[download poster PDF] 

Abstract:

Recent experiments reveal that neural populations underlying behavior reorganize their tunings over days to weeks, even for routine tasks. How can we reconcile stable behavioral performance with ongoing reconfiguration in the underlying neural populations? We examine drift in the population encoding of learned behaviour in posterior parietal cortex of mice navigating a virtual-reality maze environment. Over five to seven days, we find a subspace of population activity that can partially decode behaviour despite shifts in single-neuron tunings. Additionally, directions of trial-to-trial variability on a single day predict the direction of drift observed on the following day. We conclude that day-to-day drift is concentrated in a subspace that could facilitate stable decoding if trial-to-trial variability lies in an encoding-null space. However, a residual component of drift remains aligned with the task-coding subspace, eventually disrupting a fixed decoder on longer timescales. We illustrate that this slower drift could be compensated in a biologically plausible way, with minimal synaptic weight changes and using a weak error signal. We conjecture that behavioral stability is achieved by active processes that constrain plasticity and drift to directions that preserve decoding, as well as adaptation of brain regions to ongoing changes in the neural code.

This poster can be cited as: 

Rule, M. E., Loback, A. R., Raman, D. V., Harvey, C. D., O'Leary, T. S. (2019) Constrained plasticity can compensate for ongoing drift in neural populations. [Poster] UK Neural Computation 2019, July 1st Nottingham, UK

Local learning rules to attenuate forgetting in neural networks

Another paper from our work on Restricted Boltzmann Machines (RBMs) from the Hennig lab. 

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Main points:

• We noticed that measures of synaptic importance were available from local firing statistics (at least in Boltzmann machines)
• We look at an artificial neural network that stores memories and is easy to analyze. (Hopfield nets are the zero temperature limit of a Boltzmann machine).
• We evaluated whether this local measure of synaptic importance could help stabilize important weights when networks learn multiple things that interfere with each-other
• Intuition: biological variables, like synapse size, can correlate with useful statistical quantities. This provides tricks for biologically-plausible approximations of algorithms.
• Intuition: in systems that learn, if a parameter takes on an unusual or surprising value, it is likely that this value was set through learning—and you might want to leave it fixed.

Optimal encoding in stochastic latent-variable models

Update: the review process for this was a long one, But! It is published now, in Entropy [PDF].

The sensory system encodes the external world using spiking population codes, and must contend with the fixed bandwidth and noise inherent to neural communication. In this work, we explored a machine-learning model that shares similar constraints, and examined the coding strategies it learned when trained to encode visual input.

We used Restricted Boltzmann Machines (RBMs), which are a two-layer stochastic binary neural network that learn generative models of their inputs. We explored how optimized encoders handle limited encoding resources by varying the number of binary "neurons" available to represent visual inputs.

We found several statistical signatures that emerge around an "optimal" model size: one that is just large enough to encode its inputs. Around the optimal model model size, we saw emergence of statistical features often observed in neural population codes: sparsity, decorrelation, statistical criticality, and variability suppression.

We interpret the learned encoding strategies as a way to encode stimuli with variable bit-rates over a noisy channel of fixed bandwidth. Low-noise regions of coding space are reserved for stimuli that take the most bits to describe. In our simulations, the networks learned to suppress noise by strongly silencing some neurons.

Common stimuli don't require very many bits to transmit (from a Shannon coding perspective). Such stimuli are encoded in the noisier regions of the neural code, and exhibit increased variability. Increase neural variability in the presence of limited information is another feature observed in neural population codes.

Preview: Figure 5

 

Analyses of parameter sensitivity suggests an optimal model size for encoding sensory statistics: (a) Analysis of the Fisher Information Matrix (FIM) over a range of hidden-layer sizes (top to bottom; 13 visible units). From left to right, (1) FIM eigenvalue spectra $\lambda_i$ (y-axis) over a range of inverse temperatures β indicate that model fits (β=1) past a certain size lie at a peak in their generalized susceptibility. This is a correlate of criticality in Ising spin models. Eigenvalues below $10^{−5}$ are truncated, and the largest and smallest eigenvalues are in red; (2) Important parameters in the leading FIM eigenvector align with individual hidden units, and become sparse for larger hidden layers. The eigenvector is displayed separately for the weights (matrix), and the visible (vertical) and hidden (horizontal) biases; (3) The average sensitivity of each parameter over all FIM eigenvectors, shown here as the square root of the FIM diagonal, also shows sparsity, indicating that beyond a certain size additional hidden units contribute little to model accuracy. Data is shown as in column 2; (4) Variance of the hidden unit activation as a function of stimulus energy. In larger models, units with sensitive parameters contribute to encoding low energy, less informative patterns. (b) The average sensitivity of each parameter, measured by the trace of the FIM, normalized by hidden-layer size, decreases as hidden-layer size grows. (c) Hidden unit projective fields from a model with 37 visible and 60 hidden units, ordered by relative sensitivity (rank indicated above each image). More important units (ranks 1–8) encode spatially simple features such as localized patches, while the least important ones (ranks 53–60) have complex features.

Overall, 

We showed that machine learning models that resemble spiking population codes learn coding strategies tantalizingly similar to what is seen in vivo. We also learned that the statistical features of the population code can be used to check whether a model is too small, too large, or just the right size, to encode its inputs. 

Many thanks to Martino Sorbaro and Matthias H. Hennig for sticking with this paper during the long review. This work can be cited as

Rule, M.E., Sorbaro, M. and Hennig, M.H., 2020. Optimal encoding in stochastic latent-variable Models. Entropy, 22(7), p.714.

Population coding of sensory stimuli through latent variables

Edit: This work is published now, in Entropy [PDF].

Martino Sorbaro has been doing some really interesting work exploring the encoding strategies learned by artificial neural networks. We've found similarities between the statistics of the population codes learned by Restricts Boltzmann Machines (RBMs), and those of the retina. We'll present this work as a poster at the upcoming  Integrated Systems Neuroscience in Manchester.

TL;DR:

RBMs as a model for latent-variable encoding

• Optimal latent-variable encoding of visual stimuli seems to consistently yield models near statistical criticality.  
• Poor fits (too few hidden units,under-fitting) do not exhibit this property.
• Critical RBMs mimic the retina in Zipf laws, sparsity, and decorrelation.
• Above the optimal model size, extra units are weakly constrained as measured by Fisher information.  
• Receptive fields of excess units are less retina-like.

Questions and controversy

• Is statistical criticality a general feature of factorized latent variable models?
• Is criticality in the retina expected based simply on optimal encoding?

[download poster PDF]

Abtract:

Several studies observe power-law statistics consistent with critical scaling exponents in neural data, but it is unclear whether such statistics necessarily imply criticality. In this work, we examine whether the 1/f statistics of retinal populations are inherited from visual stimuli, or whether they might emerge from collective neural dynamics independently of stimulus statistics. We examine, in silico, a latent-variable encoding model of visual scenes, and empirically explore the conditions under which such a model exhibits 1/f statistics thought to reflect criticality. Specifically, we examine the Restricted Boltzmann Machines (RBMs) as a factorized binary latent-variable model for stimulus encoding. We find two surprising results. First, latent variable models need not exhibit 1/f statistics, but that the optimal model size, reflecting the smallest model that can faithfully encode stimuli, does. We illustrate that the optimal model size can be predicted from sloppy dimensions of the Fisher information matrix (FIM), which align with a subspace spanning the superfluous latent variables. Second, the optimal-sized model can exhibit 1/f statistics even when stimuli do not, indicating that this property is not inherited from environmental statistics. Furthermore, such models exhibit properties of statistical criticality, including diverging susceptibilities. This empirical evidence suggests that 1/f statistics are neither inherited from the  environment, nor a necessary feature of accurate encoding. Rather, it suggests that parsimonious latent- variable models are naturally poised close to criticality, generating the observed 1/f statistics. Overall,  these results are consistent with conjectures in other fields that a cost-benefit trade-off between expressivity and parsimony underlies the emergence of criticality and 1/f power-law statistics. Furthermore, this works suggests that in latent-variable encoding models, the emergence of 1/f statistics reflects true criticality and is not inherited from the environmental distribution of stimuli.

The poster can be cited as:

Sorbaro, M, Rule, M., Hilgen, G., Sernagot, E. , D, Hennig, M. H. (2017) Signatures of optimal population coding of sensory stimuli through latent variables. [Poster] The second Integrated Systems Neuroscience Workshop, 7-8th September 2017, at The University of Manchester, Manchester, UK.

Edit: the paper can be cited as

Rule, M.E., Sorbaro, M. and Hennig, M.H., 2020. Optimal encoding in stochastic latent-variable Models. Entropy, 22(7), p.714.

 

Impact of fast and slow neuronal variability in output-null dimensions on motor decoding

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Output-null spaces in motor control

The idea of a null-space extends the notion of selectivity and invariance to motor cortex (Kaufman et al. 2010). Rather than asking what stimuli change the firing rate of a neuron (and what stimulus changes it is in variant to), we ask what neural activity drives movement (and what activity does not). The subspace in which neural activity in motor cortex can vary without changing behavior is called the "output-null" space.

Variables in the output-null space explains neural variability not related to an observed behavior. This residual variability may contain components related to unmeasured behavior, neural processing, and noise sources. If one has observations from behavior $X$ and output-null space $Z$, then neural covariates $Y$ are determined. Neural variability factors in to variability induced by behavior, and that induced by output-null space.