## Departmental Papers (ESE)

#### Document Type

Journal Article

#### Date of this Version

December 2003

#### Abstract

An increasing number of applications of a one-dimensional (1-D) map as an information processing element are found in the literature on artificial neural networks, image processing systems, and secure communication systems. In search of an efficient hardware implementation of a 1-D map, we discovered that the *bifurcating neuron* (BN), which was introduced elsewhere as a mathematical model of a biological neuron under the influence of an external sinusoidal signal, could provide a compact solution. The original work on the BN indicated that its firing time sequence, when it was subject to a sinusoidal driving signal, was related to the sine-circle map, suggesting that the BN can *compute* the sine-circle map. Despite its rich array of dynamical properties, the mathematical description of the BN is simple enough to lend itself to a compact circuit implementation. In this paper, we generalize the original work and show that the computational power of the BN can be extended to compute an arbitrary 1-D map. Also, we describe two possible circuit models of the BN: the programmable unijunction transistor oscillator neuron, which was introduced in the original work as a circuit model of the BN, and the integrated-circuit relaxation oscillator neuron (IRON), which was developed for more precise modeling of the BN. To demonstrate the computational power of the BN, we use the IRON to generate the bifurcation diagrams of the sine-circle map, the logistic map, as well as the tent map, and then compare them with exact numerical versions. The programming of the BN to compute an arbitrary map can be done simply by changing the waveform of the driving signal, which is given to the BN externally; this feature makes the circuit models of the BN especially useful in the circuit implementation of a network of 1-D maps.

#### Keywords

Bifurcating neuron (BN), coupled map lattice, neural network, one-dimensional (1-D) map, parametrically coupled map lattice (PCML)

#### Recommended Citation

Emilio Del Moral Hernandez, Geehyuk Lee, and Nabil H. Farhat, "Analog Realization of Arbitrary One-Dimensional Maps", . December 2003.

**Date Posted:** 09 November 2004

This document has been peer reviewed.

## Comments

Copyright 2003 IEEE. Reprinted from

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Volume 50, Issue 12, December 2003, pages 1538-1547.Publisher URL: http://ieeexplore.ieee.org/xpl/tocresult.jsp?isNumber=28126&puNumber=81

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