Interdisciplinarity

Fun with reverse-caption images

Our department asked for an image to accompany a blog post on a manuscript we published recently. I explored using a reverse-caption image generation bot based on vqgan/clip to generate some (github here). Perhaps for the best, colleagues voted against using them. Here they are for posterity, and their accompanying captions.

paint an anatomical diagram of the brain on the left, which disassembles itself into Mongolian script in the middle, then reassembles itself into an abstract geometric object in the style of Kandinsky on the right.

An ODE with smooth, bounded solutions that computes a discontinous function in finite time

I'd like to share an Ordinary Differential Equation (ODE) that I find entertaining.

For initial condition $x(0) = x_0 \in \mathbb R$, the following ODE converges to $\operatorname{sign}(x_0)$ in finite time: \begin{equation}\begin{aligned} \tau \dot x &= \operatorname{sq}\{ x - x^3 \}, \end{aligned}\tag{1}\end{equation} where $\tau$ is a time constant controlling how quickly the ODE evolves and $\operatorname {sq}(\cdot)$ is the signed square root function $\operatorname {sq}(a) = \operatorname{sign(a)}\cdot\sqrt{|a|}$ . If we set the time constant to $\tau = {\sqrt{8\pi}}/{\Gamma(\tfrac 1 4)^2}$, we find that all $x_0$ reach $\operatorname{sign}(x_0)$ by time $t=1$.

Figure 1a shows the vector field for Equation $(1)$. Note that it's slope is vertical (infinite) at $x\in\{-1,0,1\}$ and so the vector field is not Lipschitz continuous. This is required for finite-time convergence, but removes the uniqueness of solutions. In particular, you cannot continue the final-value problem back in time from $x_f\in\{-1,0,1\}$.

Derivation from polar form

Equation $(1)$ can be derived from the following ODE on a circular variable $\theta$ \begin{equation}\begin{aligned} \dot\theta &= \tfrac 1 2 \operatorname{sq}\{ \sin(2\theta) \}, \end{aligned}\tag{2}\end{equation} where $\theta\in[0,\pi)$ and $x\in\mathbb R$. Equation $(2)$ will pull $\theta$ toward $\theta\operatorname{mod}\pi = \pm\pi/2$, or leave it stationary at the unstable fixed points $\theta\operatorname{mod}\pi=0$. We apply the change of variables $x = \tan(\frac\theta 2)$ to Equation $(2)$. Note that $\theta = 2 \tan^{-1}(x)$ and $\frac{dx}{d\theta} = \frac 1 2 (1+x^2)$. This yields: $$ \dot x = \tfrac 1 4 (1 + x^2)\, \operatorname{sq}\{ 4 \tfrac{x-x^3}{(1 + x^2)^2} \}. $$ Most terms cancel if we expand the signed square root: $$ \dot x = \tfrac 1 4 (1 + x^2) \sqrt{\left\lvert4 \tfrac{x-x^3}{(1 + x^2)^2}\right\rvert} \cdot\operatorname{sign}\big\{ 4 \tfrac{x-x^3}{(1 + x^2)^2}\big\}, $$ and we recover Equation $(1)$ if we cancel the $1+x^2$ term and drop $4/(1+x^2)^2$ from inside the $\operatorname{sign}(\cdot)$ term (since it is always positive): \begin{equation}\begin{aligned} \dot x &= \sqrt{\left\lvert {x-x^3}\right\rvert} \cdot\operatorname{sign}\{ x-x^3 \} =\operatorname{sq}\{ x - x^3 \}. \end{aligned}\end{equation}

The peculiar time constant $\tau = {\sqrt{8\pi}}/{\Gamma(\tfrac 1 4)^2}$ can be solved as the maximum convergence time for the polar form $(2)$ using Elliptic integrals.

Remarks

This ODE is notable in that the mapping $x_0 \mapsto x(1) = \operatorname{sign}(x_0)$ computes a discontinuous function in finite time, with all solutions remaining smooth. It also misbehaves in some other interesting ways and is a nice pathological example of what can happen if your vector field isn't Lipschitz. My main motivation for deriving this was to find an ODE whose solutions are smooth and bounded, but for which backpropagation through time fails due to gradients vanishing.

Self-Healing Neural Codes

A first-draft of a new manuscript is up on bioRχiv This wraps up some loose-ends from our previous work, which examined how the brain might use constantly shifting neural representations in sensorimotor tasks. 

In "Self-Healing Neural Codes", we show that modelling experiments predict that homeostatic mechanisms could help the brain maintain consistent interpretations of shifting neural representations. This could allow for internal representations to be continuously re-consolidated, and allow the brain to reconfigure how single neurons are used without forgetting. 

Here's the abstract: 

Recently, we proposed that neurons might continuously exchange prediction error signals in order to support ``coordinated drift''. In coordinated drift, neurons track unstable population codes by updating how they read-out population activity. In this work, we show how coordinated drift might be achieved without a reward-driven error signal in a semi-supervised way, using redundant population representations. We discuss scenarios in which an error-correcting code might be combined with neural plasticity to ensure long-lived representations despite drift, which we call ``self-healing codes''. Self-healing codes imply three signatures of population activity that we see in vivo (1) Low-dimensional manifold activity; (2) Neural representations that reconfigure while preserving the code geometry; and (3) Neuronal tuning fading in and out, or changing preferred tuning abruptly to a new location. We also show that additional mechanisms, like population response normalization and recurrent predictive computations, stabilize codes further. These results are consistent with long-term neural recordings of representational drift in both hippocampus and posterior parietal cortex. The model we explore here outlines neurally plausible mechanisms for long-term stable readouts from drifting population codes, as well as explaining some features of the statistics of drift.

Feedback welcome ( :

 

The Information Theory of Developmental Pruning: Optimizing Global Network Architecture Using Local Synaptic Rules

Another paper from the Hennig lab is out, this one is from Carolin Scholl's master's thesis. Once again, we used an artificial neural network to get intuition about biology. The paper is on BioRiv, and you can also get the PDF here. 

Brain–Machine Interfaces: Closed-Loop Control in an Adaptive System

Edit: I'm pleased to announce that the in-press preprint is now available from Annual Reviews [pdf].

During the first pandemic lockdown in 2020, I had the pleasure of preparing an introductory review on brain-machine interfaces with Ethan Sorrell. It will be published in the 2021 Annual Review of Control, Robotics, and Autonomous Systems. The review is in press now, but I thought I'd share a little sneak peak by way of some figures.

More figures and clip-art are on github. The clip art and figure components are free to reuse (CC NY-NC 4), but Annual Reviews owns the copyright to composed figures and sub-figures.

Convolution with the Hartley transform

The Hartley transform can be computed by summing the real and imaginary parts of the Fourier transform.

\begin{equation}\begin{aligned} \mathcal F \mathbf a &= \mathbf x + i\mathbf y \\ \mathcal H \mathbf a &= \mathbf x + \mathbf y, \end{aligned}\end{equation}

where $\mathbf a$, $\mathbf x$, and $\mathbf y$ are real-valued vectors, $\mathcal F$ is the Fourier transform, and $\mathcal H$ is the Hartley transform. It has several useful properties.

• It is unitary, and also an involution: it is its own inverse.
• Its output is real-valued, so it can be used with numerical routines that cannot handle complex numbers.
• It can be computed in $\mathcal O (n \log(n))$ time using standard Fast Fourier Transform (FFT) libraries.