Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analysing Rescaling, Discretization, and Linearization in RNNs for Neural System Modelling

Mariano Caruso, Cecilia Jarne

TL;DR

The paper addresses how temporal rescaling, discretisation, and linearisation interact when applied to continuous-time RNNs used to model neural dynamics. It provides a rigorous mathematical analysis, deriving transformed equations under each operation and identifying precise commutativity criteria. Key contributions include that rescaling and discretisation commute when time-step adjustments align with the scaling, and that linearisation commutes with discretisation or rescaling near equilibrium, along with explicit expressions and stability implications for biologically plausible neural dynamics. This framework offers a robust, flexible guide for implementing RNNs in neuroscience, balancing analytical tractability with biological realism in neural dynamics simulations.

Abstract

Recurrent Neural Networks (RNNs) are widely used for modelling neural activity, yet the mathematical interplay of core procedures is used to analyze them (temporal rescaling, discretization, and linearization) remain uncharacterized. This study establishes the conditions under which these procedures commute, enabling flexible application in computational neuroscience. We rigorously analyze the mathematical foundations of the three procedures, formalizing their application to continuous-time RNN dynamics governed by differential equations. By deriving transformed equations under rescaling, discretization, and linearization, we determine commutativity criteria and evaluate their effects on network stability, numerical implementation, and linear approximations. We demonstrate that rescaling and discretization commute when time-step adjustments align with scaling factors. Similarly, linearization and discretization (or rescaling) yield equivalent dynamics regardless of order, provided activation functions operate near equilibrium points. Our findings directly guide the design of biologically plausible RNNs for simulating neural dynamics in decision-making and motor control, where temporal alignment and stability are critical

Analysing Rescaling, Discretization, and Linearization in RNNs for Neural System Modelling

TL;DR

The paper addresses how temporal rescaling, discretisation, and linearisation interact when applied to continuous-time RNNs used to model neural dynamics. It provides a rigorous mathematical analysis, deriving transformed equations under each operation and identifying precise commutativity criteria. Key contributions include that rescaling and discretisation commute when time-step adjustments align with the scaling, and that linearisation commutes with discretisation or rescaling near equilibrium, along with explicit expressions and stability implications for biologically plausible neural dynamics. This framework offers a robust, flexible guide for implementing RNNs in neuroscience, balancing analytical tractability with biological realism in neural dynamics simulations.

Abstract

Recurrent Neural Networks (RNNs) are widely used for modelling neural activity, yet the mathematical interplay of core procedures is used to analyze them (temporal rescaling, discretization, and linearization) remain uncharacterized. This study establishes the conditions under which these procedures commute, enabling flexible application in computational neuroscience. We rigorously analyze the mathematical foundations of the three procedures, formalizing their application to continuous-time RNN dynamics governed by differential equations. By deriving transformed equations under rescaling, discretization, and linearization, we determine commutativity criteria and evaluate their effects on network stability, numerical implementation, and linear approximations. We demonstrate that rescaling and discretization commute when time-step adjustments align with scaling factors. Similarly, linearization and discretization (or rescaling) yield equivalent dynamics regardless of order, provided activation functions operate near equilibrium points. Our findings directly guide the design of biologically plausible RNNs for simulating neural dynamics in decision-making and motor control, where temporal alignment and stability are critical
Paper Structure (6 sections, 13 equations, 1 figure)

This paper contains 6 sections, 13 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Introduction
  3. Rescaling
  4. Discretisation
  5. Linearisation
  6. Discussion

Figures (1)

  • Figure 1: Left panel: Schema of how a RNN represent a brain cognitive area. Operations over the RNN. a) Scheme of how the equations are modified via the process of temporal rescaling and discretization. b)Linearisation and Discretisation process implemented in the equation for the activity.c)Linearisation and Rescaling procedures implemented on the activity of units.