Input-driven circuit reconfiguration in critical recurrent neural networks

TL;DR

The paper addresses rapid input-driven reconfiguration of neural circuits without changing synaptic weights, enabling context-dependent computation such as connectedness detection. It proposes a minimal single-layer recurrent convolutional network with a unitary kernel $U$ and complex-valued state $Z$, updated by $Z_{n+1}=\phi(U\otimes Z_n + I_n)$ with the activation $\phi(z)=\frac{z}{\sqrt{1+|z|^2}}$, and analyzes its critical dynamics. By patterning the input $I_0$ and using the gain field $\Gamma=\phi'(U\otimes Z^*+I_0)$, the model creates regions where traveling waves propagate or attenuate, implementing a geometric IF statement that enables floodfill and related routing. The work connects continuous- and discrete-time perspectives via exponentiation of generators, discusses Fourier-domain representations of unitary couplings, and demonstrates implications for brain-inspired reconfigurable computation and neuromorphic design.

Abstract

Changing a circuit dynamically, without actually changing the hardware itself, is called reconfiguration, and is of great importance due to its manifold technological applications. Circuit reconfiguration appears to be a feature of the cerebral cortex, and hence understanding the neuroarchitectural and dynamical features underlying self-reconfiguration may prove key to elucidate brain function. We present a very simple single-layer recurrent network, whose signal pathways can be reconfigured "on the fly" using only its inputs, with no changes to its synaptic weights. We use the low spatio-temporal frequencies of the input to landscape the ongoing activity, which in turn permits or denies the propagation of traveling waves. This mechanism uses the inherent properties of dynamically-critical systems, which we guarantee through unitary convolution kernels. We show this network solves the classical connectedness problem, by allowing signal propagation only along the regions to be evaluated for connectedness and forbidding it elsewhere.

Figures (5)

• Figure 1: From chandramm. The fixed point $z_{*}=\phi(z_{*}+I)$ with $\phi(z)=z/\sqrt{1+z^{2}}$; $z_{*}$ as a function of $I$ (red), together with $\phi'(z_{*})=\left(1+z_{*}^{2}\right){}^{-\frac{2}{3}}$, the slope of $\phi$ at the fixed point controlling decay rate (blue), and $-1/\log(\phi'(z_{*}))$ (black), the relaxation time to the fixed point. The thin lines are power-law asymptotics to the low $I$ regime: $z_{*}\approx\left(2I\right)^{\frac{{1}}{3}}$ and $\tau\approx\frac{2}{3}\left(2I\right)^{-\frac{{2}}{3}}$.
• Figure 2: 1D network (N=2048). Top panel. $\Gamma$ (top line, black) was chosen to have 8 alternating cycles between $\Gamma=1$ and $\Gamma=\Gamma_{0}$. From there $Z_{*}$ is computed (green line) and then $I_{0}$ (real part, red; imaginary part, blue). Bottom panel: An oscillatory signal is injected at the center (a non-attenuating region) and propagates outwards. Notice it exponentially decays as it traverses $\Gamma<1$ areas and does not attenuate through $\Gamma=1$ areas.
• Figure 3: Two vertically-stacked boxes. The gray areas cannot be traversed. In addition to the input $I_{0}$ patterning the boxes, an additional oscillatory input is applied at the center of the top box. Left, the middle wall is intact; the signal injected at the top box stays in the top box. Center, a small aperture is broken in the middle wall; the signal leaks into the bottom box. Full movies in Supp Mat. This demonstrates the walls act as a geometric IF statement: something is allowed to happen or not depending on an input.
• Figure 4: A labyrinthine pattern was created using band-passed noise to pattern areas where propagation is allowed ($\Gamma=1$, in black) or strongly disallowed ($\Gamma=0.01$, gray). An oscillating signal at the frequency of an eigenvalue was then injected at a single pixel the center of the figure; the network was evolved and this figure shows $\log(|Z|)$ in color code. $Z$ is a 2048x2048 long complex array; $U$ is the exponential of a numerical Laplacian kernel (times $i$). Please notice that the oscillatory signal did not fill non-contiguous areas, implementing the classic floodfill algorithm (a.k.a. paint bucket). Full movies are in Supplementary Materials
• Figure 5: Lighthouse.