Hierarchically Modular Dynamical Neural Network Relaxing in a Warped Space: Basic Model and its Characteristics

TL;DR

This paper introduces a hierarchically modular dynamical neural network model that relaxes in a warped space, capable of learning and associating signals with complex, curved convergence trajectories based on internal and external space interactions.

Contribution

It presents a novel neural network framework combining static and dynamic neurons with layered internetworks, enabling complex signal mapping and trajectory generation in a warped space.

Findings

The model can form 2D mappings using sinusoidal signals similar to Lissajous curves.

Convergence speed varies with trained internetwork mappings, affecting trajectory curvature.

The system can generate output trajectories mapped inversely from the external to internal space.

Abstract

We propose a hierarchically modular, dynamical neural network model whose architecture minimizes a specifically designed energy function and defines its temporal characteristics. The model has an internal and an external space that are connected with a layered internetwork that consists of a pair of forward and backward subnets composed of static neurons (with an instantaneous time-course). Dynamical neurons with large time constants in the internal space determine the overall time-course. The model offers a framework in which state variables in the network relax in a warped space, due to the cooperation between dynamic and static neurons. We assume that the system operates in either a learning or an association mode, depending on the presence or absence of feedback paths and input ports. In the learning mode, synaptic weights in the internetwork are modified by strong inputs corresponding to repetitive neuronal bursting, which represents sinusoidal or quasi-sinusoidal waves in the short-term average density of nerve impulses or in the membrane potential. A two-dimensional mapping relationship can be formed by employing signals with different frequencies based on the same mechanism as Lissajous curves. In the association mode, the speed of convergence to a goal point greatly varies with the mapping relationship of the previously trained internetwork, and owing to this property, the convergence trajectory in the two-dimensional model with the non-linear mapping internetwork cannot go straight but instead must curve. We further introduce a constrained association mode with a given target trajectory and elucidate that in the internal space, an output trajectory is generated, which is mapped from the external space according to the inverse of the mapping relationship of the forward subnet.