Temporal Modeling of Neural Net Input/Output Behaviors: The Case of XOR

Bernard Zeigler,Alexandre Muzy

doi:10.3390/systems5010007

Abstract

In the context of the modeling and simulation of neural nets, we formulate definitions for the behavioral realization of memoryless functions. The definitions of realization are substantively different for deterministic and stochastic systems constructed of neuron-inspired components. In contrast to earlier generations of neural net models, third generation spiking neural nets exhibit important temporal and dynamic properties, and random neural nets provide alternative probabilistic approaches. Our definitions of realization are based on the Discrete Event System Specification (DEVS) formalism that fundamentally include temporal and probabilistic characteristics of neuron system inputs, state, and outputs. The realizations that we construct—in particular for the Exclusive Or (XOR) logic gate—provide insight into the temporal and probabilistic characteristics that real neural systems might display. Our results provide a solid system-theoretical foundation and simulation modeling framework for the high-performance computational support of such applications.

Highlights

Bridging the gap between neural circuits and overall behavior is facilitated by an intermediate level of neural computations that occur in individual and populations of neurons [1]
The computations performed by Artificial Neural Nets (ANN) can be viewed as a very special, but currently popular, instantiation of such a concept [2]
Having a somewhat formal definition of what it means for a discrete event model to display a behavior equivalent to computing a memoryless function, we turn toward discussing Discrete Event System Specification (DEVS) models that can exhibit such behaviors for the XOR function

Summary

Introduction

Bridging the gap between neural circuits and overall behavior is facilitated by an intermediate level of neural computations that occur in individual and populations of neurons [1]. We review system specifications and concepts for their input/output (I/O) behaviors that allow us to provide definitions for systems implementation of memoryless functions This allows us to consider the temporal characteristics of neural nets in relation to the functions they implement. This example shows that even for a simple function, such as adding two values, there can be considerable complexity involved in the specification of behavior when the temporal pattern of the messages bearing such values is considered.

Systems Implementation of a Memoryless Function

DEVS Deterministic Representation of Gelenbe Neuron

Realization of the XOR Function

Probabilistic System Implementation of XOR

Discussion