Excitatory-Inhibitory Recurrent Dynamics Produce Robust Visual Grids and Stable Attractors

Abstract

Spatially modulated grid cells has been recently found in the rat secondary visual cortex (V2) during activation navigation. However, the computational mechanism and functional significance of V2 grid cells remain unknown, and a theory-driven conceptual model for experimentally observed visual grids is missing. To address the knowledge gap and make experimentally testable predictions, here we trained a biologically-inspired excitatory-inhibitory recurrent neural network (E/I-RNN) to perform a two-dimensional spatial navigation task with multisensory (e.g., velocity, acceleration, and visual) input. We found grid-like responses in both excitatory and inhibitory RNN units, and these grid responses were robust with respect to the choices of spatial cues, dimensionality of visual input, activation function, and network connectivity. Dimensionality reduction analysis of population responses revealed a low-dimensional torus-like manifold and attractor, showing the stability of grid patterns with respect to new visual input, new trajectory and relative speed. We found that functionally similar receptive fields with strong excitatory-to-excitatory connection appeared within fully connected as well as structurally connected networks, suggesting a link between functional grid clusters and structural network. Additionally, multistable torus-like attractors emerged with increasing sparsity in inter- and intra-subnetwork connectivity. Finally, irregular grid patterns were found in a convolutional neural network (CNN)-RNN architecture while performing a visual sequence recognition task. Together, our results suggest new computational mechanisms of V2 grid cells in both spatial and non-spatial tasks.

Mediodorsal thalamus regulates sensory and mapping uncertainties in flexible decision making

“…Fixed-point analysis. Similar to our previous analysis 21,22 , we identified fixed-points or slow points of the performance-optimized PFC-MD model by numerically solving the optimization problem (https://github.com/mattgolub/fixed-point-finder) min 𝐱 j 𝑞(𝐱 j), where 𝑞(𝐱 j) = l−𝐱 j + 𝐖 ,== 𝜙(𝐱 j) + [𝐖 /0 , 𝐕 /0 ] • 𝐮l p (which was not certified by peer review) is the author/funder. All rights reserved.…”

“…We first constructed a PFC-alone network as the baseline model. The PFC network is an excitatory-inhibitory (E/I) recurrent neural network (RNN) with N PFC =256 fully interconnected units described by a standard firing-rate model 20,22 . We adopted a continuous-time formulation of RNN dynamics as follows: where τ denotes the time constant (we used τ=20 ms), ξ denotes additive N PFC -dimensional Gaussian noise, each independently drawn from a standard normal distribution, σ defines the scale of the noise standard deviation; W rec is an N PFC × N PFC matrix of recurrent connection weights, and W in denotes an N PFC × N in matrix of connection weights from the input to network units ( N in =72).…”

“…Similar to our previous analysis 21,22 , we identified fixed-points or slow points of the performance-optimized PFC-MD model by numerically solving the optimization problem (https://github.com/mattgolub/fixed-point-finder) where . We collected a set of fixed-points by randomly initializing the network.…”

“…A square matrix A is non-normal if it satisfies AA T ≠ A T A. For the non-normal matrix W eff (in the PFC-MD model) or W rec (in the MD-lesioned model), we conducted eigenvalue analysis to analyze the nonlinear dynamics with linear approximation 22,71 , where the maximum of the real (or real-part) eigenvalues characterizes the long-term behavior for the speed of network’s steady state decaying to zero 72 . We further we adapted a published method to estimate the intrinsic time constant of dynamical systems 73 , for both PFC-MD and MD-lesioned structures ( Supplementary Fig.…”

“…We first constructed a PFC-alone network as the baseline model. The PFC network is an excitatory-inhibitory (E/I) recurrent neural network (RNN) with NPFC=256 fully interconnected units described by a standard firing-rate model 20,22 . We adopted a continuous-time formulation of RNN dynamics as follows:…”