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CIRCUIT IMPLEMENTATION

OF

NEURA L FITZHUGH NAGUMO EQUA TIONS

Edgar Shchee-Sin encio and Bernab6 Linares-Barranco

Texas A M University

Department of Electrical Engineering

College Station, Texas

77843-3128

Abstract:

A circuit is proposed that emulates FitzHugh-

Nagumos differential equations using OTAs, diodes and ca-

pacitors. These equations model the fundamental behavior

of an organic neuron cell, i.e., if the input is below a certa in

threshold, no output is observed, and if it

is

above threshold,

the output yields a sequence of k in g pulses. The circuit is ob-

tained by using a general method for implementing nonlinear

differential equations. The resulting circuit due to the (voltage)

programmability of the OTA allows one to easily vary param-

eters. Thus a large family

of

solutions can be obtained includ-

ing the Van der Pols equation. Experimental results from a

breadboard prototype are given that show the suitability of the

technique used and their potential for CMOS implementation.

Introduction

Different types of artificial neurons have been used until

now which are usually inspired in the biological neural cells.

Most of these artificial neurons are extremely simplified ver-

sions of the real ones, such

as

those used by Hopfield (1-41,

Anderson

[5],

Rumelhart

(61,

Kohonen

[7],

and in most of

the papers in the expanding neural literature. These conven-

tional neuron models [I-71 can be classified as non-oscillatory

neurons. There has also been attempts to model the oscilla-

tory nature

of

the biological neurons using hysteretic devices

[8-lo].

These models are supposed to have a similar behav-

ior to the more detailed modeling of FitzHugh-Nagumo

Ill]

which is a simplified version of one

of

the most exact model

due to .Hodgkin and Huxley

(121.

In this paper, we propose

a hardware implementation of the FitzHugh-Nagumos equa-

tions with techniques suitable for CMOS integration. The de-

@ L4

Fig. 1. General Topology

tial Equations

Representing N Nonlinear Differen-

vices used are operational transconductance amplifiers (OTAs)

(131,

capacitors and diodes. In the next section we give a

methodology for implementing a general set of fi rst order non-

linear differential equations using OTAs. As a particular case,

a circuit th at solves FitzHugh-Nagumos equations is intro-

duced and experimental results of a protoboard real ization will

be given. A CMOS prototype has been sent for fabrication in

a 2pm p-well, double-metal, double-poly process.

Implementing Nonlinear Differential Equations

A general circuit topology that solves a set of

N

nonlinear

differential equations with N variables is shown in Fig.

1.

Each node Vj is connected to each other node V; hrough

a capacitor Ci,, and to ground through Cj,. Two current

sourcesI L ~nd

N^

are also connected to node

j.

IL

s linearly

dependent on th e node voltages of all the other nodes

where gmj (which can be positive or negative) is the transcon-

ductance relating interaction between nodesm and j and Ioj s

an independent current source representing an external input.

I N j is a nonlinearly dependent current source

For each node

Vj

the following

KCL

equation holds:

by defining

, i f i j

(3) can be rewritten as

Here e

89CH2785-4i901WBO2444/90/000M)244S01.00990

IEEE

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and in matrix notation

5)

becomes

IF

GVT fT V) BVT 0

6)

This is the general set of

N

nonlinear differential equations.

FitzHugh-Nagumo's equations represent a second order system

and are a particular case of

6) ll]

where it has been made

g12 = gml, gms = -gll

and

gma=

-921

fl -) = 0

and

f2 .)

=

-f(Vz). f .)

is an N-shaped

nonlinear function that will be shown later, and

gmi, Ci,

are

positive parameters.

Circuit

Implementation

tions

7).

The circuit of Fig.

2

simulates the set of differential equa-

Fig.

2

Circuit Implementation of FitzHugh-Nagumo's Equa-

The exact form of the function

f .)

does not seem to be

very critical. Originally

[11]

cubic polynomial was suggested,

but a piecewise linear dependance can give the same basic

properties to the system

[14].

We will consider this latter ap-

proach, f .) being

as

shown in Fig.

3.

tion.

Fig. 3 N-Shaped Transfer Function of the Nonlinear Circuit

A phase portrai t of the equilibrium points of the system

Element.

described by

7)

is shown in Fig.

4.

I

I I

Fig.

4

Phase Portrait

of the

System Characterized by 7).

The equilibrium points are obtained when VI

=

V = 0.

Since we have a nonlinearity with th ree linear segments, we

can divide the phase plane in three linear regions. Namely,

regions Q 8 and as shown in Fig.

4.

For region

Q

the

state equations are given by

The equilibrium point A (real)' for this region is stable

[15]

f

the trace and determinant of the system satisfy tha t

T &- ?>O

a gmlgmz 9a9m3

9)

z z c 1 1

CllC22 CllC22

> O

For regions Q and Q we can describe the behavior of the

system, the and signs in (10) correspond to the regions

Q and

0

y

-9ma 3

C z a

C a a

C z a

and their equilibrium points

B

and C, respectively, are stable

1151

if

Virtual equilibrium point is the one that is outside the region

in which the steady state solution is defined.

Observe that points B and

C

as shown in Fig.

4

are vir-

tual points: while the system is working in regions

Q

or

@

it

will be attracted by the stable points B and C, respectively.

But

as

it approaches those points, the system will enter region

0 where the equilibrium point

A

is unstable and now the sys-

tem will try to leave region Q. We can intuitively see that

some limit cycle will be reached where the system will remain

oscillating

[ll].

t is also simple to see from Fig.

4

hat

if

the

relative positions of the curves

VI = 0

and

Vz = 0

are changed,

by varying Iol and/or

Io',

one or both of the points

B

and

C

can become a real point, instead of a vir tual one.

In

this

case the system will reach this point and stay there, th at

is,

no

oscillations will be produced.

For the nonlinear resistor of Fig.

2,

the technique pro-

posed in

[16]

s used, that implements nonlinear I/V transfer

characteristics with OTAs and

MOS

diodes. The resulting cir-

cuit for this element is shown in Fig.

5.

-f

4 b

Fig.

5.

Implementation of the Nonlinear Function Using OTAs

and Diodes.

B or c will become real points

if

they are located in regions @ or

0

espectively. Real equilibrium point is the one that belongs to the

region in which the steady state solution is defined.

45

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Theor

et c a1 Simulations

In order to have a good reference about our next experi-

mental results, we first show a mathematical simulation of the

set

of

differentia l equations (7a) and (7b) using the piece-wise

linear N-shaped nonlinear transfer function f V z ) hown in

Fig. 3.The simulation was performed using the circuit simu-

lator SPIC E with ideal transconductance amplifiers and ideal

diodes. In Fig.

6

is shown the theoretical results when mak-

ing I,1 =

I z

= 0

in

Fig. 2 i.e., the case of self-oscillatory

output . As we will show in the next sec tion the shape of wave-

forms VI t )and Vz t ) f Fig. 6are the same

as

hose measured

experiment ally.

Fig. 6 SPI CE Simulation of (7) when I,, =

I =

0 (Self-

Oscillatory).

Ex p er imen ta l Resu l ts

A breadboard prototype has been built with discrete bipo-

lar commercially available components, in order to verify th e

expected results. The OTA used has been the LM 13600 [13]

and two 10% tolerance ceramic capacitors were used with a

value of 270 pF. In order to obtain the FitzHugh-Nagumo os-

cillations of a neuron it is required that the time constant of

eq. (7a) to be much smaller t han th e one associated with (7b).

Since both capacitors are equal, we just simply make

Q m l ,

Qm3

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Fig. 11

Nonlinear Transfer Characteristics of Fig.

5.

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