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

Gaussian moment-closures for autoregressive GLM population models

[get notes as PDF]

In Rule and Sanguinetti (2008) "Autoregressive Point Processes as Latent State-Space Models" (more at the blog post) we developed moment-closure approximations for the time-evolution of the moments of Autoregressive Point-Process Generalized Linear Models (AR-PPGLMs) of neural activity. These notes generalize these approximations for AR-PPGLM models of neural ensembles . For networks of linear-nonlinear "neurons", the derivations are very similar to the single-neuron case presented in the paper.

Inferring unobserved neural field intensities from spiking observations

Edit: I am very happy to report that this work has now been published in PLoS Computational Biology.

I'll be presenting our ongoing work on merging neural field models with statistical inference at the Integrated Systems Neuroscience Workshop in Manchester, and at the Bernstein Conference in Göttingen. [get poster PDF]

What's exciting about this work is that it combines modelling principles from statistical physics and statistical inference. We start with a detailed microscopic model, and then construct a second-order neural field model, which is then used directly for statistical inference. Normally, neural field models are only treated as abstract, qualitative mathematical models, and are rarely integrated with data. 

Video: Simulation of 3-state Quiescent-Active-Refractory blue, red, green) neural field model of spontaneous retinal waves that occur during development. Waves are generated by the inner retina, and drive retinal ganglion cell spiking, which we can observe on a high-density multi-electrode array. [get original avi from Github]

Directional statistics for spatiotemporal wave analysis

I've been searching for a good distribution that can be used to summarize how the distribution of phases and amplitudes evolves during transient synchronization events of beta (~20 Hz) local field potentials (LFP) in motor cortex. So far, it seems difficult to find a single distribution family that works in all cases. 

Spatiotemporal wave activity in beta oscillations in motor cortex can be described in terms of the beta-band analytic LFP signal, which has both a magnitude and a phase, and who's real part is equal to the time-domain value of the beta-filtered signal. 

\begin{equation}z_k(t) = \beta_k (t) + i\cdot  \operatorname{Hilbert}(\beta_k)(t) = r_k(t) e^{i \theta_k(t)},\\\textrm{ where $k$ indexes over channels}\end{equation}

Circular statistics can be used to summarize the distribution of analytic signal phase, in order to detect synchrony and wave events.

[read more in the PDF

Figure 2: Neither the complex Gaussian nor log-polar statistics perfectly describe the distributions of analytic signal. In these plots, the black ellipse represents a complex Gaussian model of the data, with the ellipse boundary at one standard deviation, and the ellipse axes representing the eigenvectors of the covariance matrix $\Sigma$. The cyan contours represent a log-polar model of the data, which uses the mean and standard deviation of the log-amplitude, as well as the circular mean and standard deviation of the phases, to model the data in log-polar space. (a) When phase is concentrated, and not correlated with amplitude, both the log-polar statistics and the complex Gaussian distribution describe the data well. (b) When phase and amplitude are correlated, the log-polar model cannot capture the phase-amplitude dependence. (c) During traveling wave events, signal amplitude is high, and there is dispersion in phase. In these cases, the log-polar statistics are more appropriate than the complex Gaussian. (d) Traveling wave events appear to often evolve from states that show a mixture of synchrony and standing wave dynamics. The log-polar statistics break down when the phase distribution is bimodal, but the complex Gaussian can describe these states well. (e) At low signal amplitudes, the system is often asynchronous, and the phase and amplitude of the log-polar model are poorly defined. (f) Although rare or absent in our data, a hypothetical distribution with uniform phase and concentrated amplitude could occur, say, during traveling wave events with short wavelength. In this case, the complex Gaussian model is especially bad.

A mechanism for persistent memory in oscillatory networks

I just finished the Methods in Computational Neuroscience "bootcamp". It was intense, but valuable—I can recommend that any student/postdoc who wants to get started in the field take it. 

For a course project, I extended previous work, which showed that oscillatory drive can excite patterned instabilities in visual cortex. These patterns constitute attractor states that are stabilized by an external oscillatory drive. The project explored the idea that similar oscillation-stabilized attractors might serve as a flexible working memory. 

(Unfortunately, these notes are rather brief and incomplete as I never had the time to write this up properly. )

Selective, short-term maintenance of attractor states is important for working memory in neural systems. Neural networks, however, are typically stabilized by feedback inhibition. The delays in this inhibition lead to unavoidable oscillations. 

This project demonstrate a scenario where the attractor dynamics associated with a working memory are not fixed points, but instead fixed limit cycles embedded within a population oscillation. 

External oscillatory drive, perhaps e.g. an attention signal, can switch such a network from a "read-in" mode, in which the system responds to external input, and a "hold" model, in which the system retains a memory of its past states. These concepts may underlie flexible working memory solutions that can coexist with population oscillations.

To better understand the mechanisms underlying this, we explored a switched piece-wise linear model that captured the qualitative dynamics of the original system. We found that firing-rate nonlinearities with positive curvature are important for allowing a synchronous (non-selective) external signal to excite the asymmetric network states associated with a memory trace. 

[get writeup PDF]

[get presentation slides]

Preview: Figure 4

Figure 4: Many-population generalization of the two-population model of Rule et al. (2011), serving as a working memory. Starting from rest, a stimulus is delivered to the first oscillator (green) for $t\in[300,500)$ ms. We test the stability of this stimulus-driven network response during a hold period $t\in[500,1500)$ ms. Without oscillatory drive (top), the memory fades in a few cycles. Periodic stimulation (bottom) preserves the memory. Rightmost plots show the phase-plane dynamics during the readout period $t\in[1000,1500)$. The top system, without drive, has returned to rest. The driven system shows a stable limit cycle, with higher firing rates in the population that initially received the stimulus.

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

[get these notes as PDF]

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.