Physica 6D (1983) 305-320 North-Holland Publishing Company

ON THE ARITHMETIC OF PHASE LOCKING: COUPLED NEURONS

AS A LATTICE ON R 2

T r a c y A L L E N * Department of Biophysics and Department of Entomology, University of California, Berkeley, USA

Received 1 December 1981 Revised manuscript received 30 August 1982

Using methods from the geometry of numbers, we derive an explicit, global solution for the phase-locking behavior of a simple integrate-and-fire model of coupled neurons. The solution gives the ratios of phase locking (rotation numbers) attained as functions of the parameters of natural frequency and bidirectional coupling. The ordering of the ratios is related to Farey-type series and to simple continued fractions. A transition between two ratios, say a/b to c/d, is possible if, and only if, ad - bc = "- I. Empirically, similar ordering is evident in published data from various neuron analogues. We compare and contrast the present results with those from models based on Caianiello's equation and on more general mappings on the torus.

"The sequence of integers might be called the prototype of a periodic phenomenon." [1]

1. Introduction

The sawtoo th funct ion of time, F ( t ) = wt

(rood I), where w is a scalar, is one of the

simplest imaginable fo rms of nonl inear oscil-

lation. Of matching simplicity is the pulsatile

form of coupl ing that we shall admit be tween

two such oscil lat ions for the purposes of this paper. As a model , this formula t ion is naively

ana logous to coupled neural pacemakers , where it is an ext reme example of an integrate-and-fire

model [2], subject to pulsatile coupling. It is also direct ly ana logous to certain electronic circuits

called time bases, which are, for example , res- ponsible for the stability of pictures on tele-

vision sets. The coupl ing here can be ei ther unidirect ional , with one device serving as input

to the other , or bidirectional , with coupl ing going both ways in varying degrees. The uni- direct ional case here is exact ly the model given by Chua [3] as a pro to typica l example of a time

* Present address: Department of Entomology, 201 Well- man Hall, University of California, Berkeley, Ca. 94720, USA.

base. The overall behav ior of even this simple

model has e luded analysis , a l though certain

connec t ions with number theory , which we

pursue here, intrigued B. van der Pol [4].

What arises f rom our formula t ion , as func-

t ions of its parameters , are zones of phase lock-

ing separated by abrupt transit ions. Figs. 7, 8

and 9 c o n v e y how these zones fit together on

the pa ramete r space. These are d rawn on the basis of our theorem 1.

Such p h e n o m e n a are typical of diverse sys- tems of coupled nonl inear oscil lators. For

example , table I shows sequences of ratios obtained empirical ly f rom (1) an electronic

neuron analogue driven by an exci tory pulse

train [5], (2) s imulat ions of a van der Pol equa- tion, with a sinusoidal forc ing term [6], (3) a pair

of isolaterally coupled tunnel diode oscil lators [7], (4) a p a c e m a k e r neuron, driven by in-

hibi tory input [9], and (5) heart pacemake r tis- sue, driven by exc i tory input [10]. In television sets, these effects can be seen by varying the horizontal and vertical "ho ld" ad jus tments , regulating the response of internal oscil lators to

0167-2789/83/0000-0000/$03.00 © 1983 Nor th -Ho l l and

306 T. A l l e n / C o u p l e d n e u r o n s as a lat t ice on R 2

synchronizat ion pulses that are received along with the picture information.

The phenomena are usually difficult to analyze mathematical ly, and the arithmetical structure implied by the sequences of ratios tends to get lost behind layers of analytical complexity. Recent advances in relation to neurons have been based on maps on the torus [2, I1, 12, 13], and on Caianiel!o's equation [14, 15, 16]. The present formulation has a spe-

cial form as a map on the torus, close to, but not within the class of maps studied by Keener [20]. However , due to combinatorical difficulties, it is usually tedious or next-to-impossible to del- ineate explicitly more than a few of the simplest zones of phase locking. The overall arithmetical structure has to be inferred by indirect means. This structure can be very complicated and difficult to visualize.

The payoff of the simple model here is that an explicit solution is easily found, using a theorem from the geometry of numbers to derive the global distribution of the phenomena over the parameter space. We show that the problem is equivalent to one of visibility in a lattice on R 2, having dimensions of time and having "opaque" triangles at the lattice points, and to which the theorem derived in [17] applies. Simple con- tinued fractions and Farey- type series provide algorithms for mapping the responses and a basis for understanding the global structure of phase locking in this system.

The final section of this paper is a discussion of this model and of other models in relation to the arithmetic of the data in table I. As an indicator of the arithmetical structure, we shall consider the "modular i ty" of neighboring

ratios. That is, if a/b and c/d are neighboring ratios, then their modulari ty equals a d - b c . A class of models, of which the present case is prototypical , predict that the ordering in the predominance of the ratios will be unimodular, a d - b c =-+ 1. The models arrive at this con- clusion in quite different ways, so we shall a t tempt to compare and contrast them to clarify

this issue. Observe that unimodular transitions are prevalent, but not universal, in the data of table I.

2. Representations of the model

The model is developed here as a geometrical idealization of an extreme integrate-and-fire (alias " threshold" or "relaxat ion") oscillator. In each oscillator of a coupled pair, some quantity

elevates linearly to a threshold, then retraces instantaneously to a baseline level, and so on in oscillation. Sans coupling, each oscillator generates a sawtooth function, F( t )= wt (mod 1). The scaling of the amplitude to unity entails no loss of generality. Two such oscillators, denote them X and Y, have independent real- valued natural frequencies, wx and Wy. There- fore, the natural f requency ratio, wy/wx ( 0 < wy/Wx < ~), is typically an irrational number.

There are various schemes by which two such oscillators could be coupled, involving various functional relations between their states and their parameters . Here we suppose that each oscillator, at each of its instants of retrace, causes an excitory pulse on the threshold of the other oscillator. See fig. 1. The magnitude of the pulse is some fraction of the baseline-to- threshold distance. The value Oxy(0~Oxy<~l )

measures the effect of X on Y, and the value ayx(0<~ a~,x~ < I) measures the effect of Y on X. The pulse is of infinitesimal width. The effect of the pulse is all-or-nothing, that is, it either resets

X time Coupling factors baseline ~ ~ , - , . x - , . ~ ~,.

: " ~ - " 0 baseline' ~ I ~ . I ~ ~ . ~ J . ~ , xy

Y time

Fig. 1. Geometrical definition of the system, showing the two "sawtooth" waveforms with their thresholds placed head-to-head. The slopes Wx and wy (= "natural" frequen- cies) and coupling magnitudes axy and ayx are normalized to a baseline-to-threshold distance of unity.

T. Al len/Coupled neurons as a lattice on R 2 307

the o p p o s i n g osc i l l a to r , or else it l eaves no

m e m o r y of its o c c u r a n c e .

This m o d e l has a P o i n c a r e ' map , wh ich g ives

the i t e ra te s of X re la t ive to phase on the cyc l e

of Y, and v isa versa . Fig. 2 shows an e x a m p l e of

the c o n s t r u c t i o n for the first i t e ra te of a map

tha t a ch i eves the ra t io n y / n x = 1/2. The map is

a l w a y s of the fo rm, 0;.~ = 0; + wy/Wx {or Wx/Wy}

(mod 1), modi f ied with a s imple m e c h a n i s m of

coupl ing , which a m o u n t s to a s egmen t of 0;+, =

1 = 0 on the s t a n d a r d unro l l ing of the torus . This

fo rm b o r d e r s on, bu t is not wi th in the c lass of

p i e c e w i s e - c o n t i n u o u s , s t r ic t ly i nc rea s ing maps

on the t o rus s tud i ed by K e e n e r [20]. I ndeed , for

the sake of an exp l i c i t so lu t ion , we look ing here

at a ve ry spec i a l i z ed mode l in the c o n t e x t of

maps on the torus .

The ra t io of phase lock ing is e q u i v a l e n t to the

ro t a t ion number . The f o r m e r is the ra t io of the

ave r age o u t p u t f r e q u e n c i e s , and the la t te r is the

ave r age phase a d v a n c e per cyc le .

W e shou ld men t ion at this po in t that the

o Oi

i- o

T o T

o 1

Fig. 2. The circle-map for a system that attains nJnx = 1/2. The map gives the phase of the i + lth iterate of the map as a function of the phase at the ith iterate. The segments "a" and "d" are parallel to the identity. The segment "b" is due to oscillator Y resetting the phase to 0 = 1 =0, and the segment "c" is due to X having the same effect. The diagram is drawn for phases of X relative to the cycle of Y. A similar map, which has one fixed point in its first iterate, can be drawn for the phases of Y relative to the cycle of X.

p r e s e n t f o rmu la t i on , as a mode l of neurons , has

an unrea l i s t i c , a l l -o r -no th ing r e s p o n s e to single

input pulses . I t is more rea l i s t ic to c o n s i d e r

mode l s in wh ich pu l ses , a d m i n i s t e r e d at va r ious

phase s of the a u t o n o m o u s cyc l e of the d r iven

osc i l l a to r , c ause g r aded a d v a n c e s or d e l a y s in

the phase of the d r iven osc i l la tor . This is a

c h a r a c t e r i s t i c of the H o d g k i n - H u x l e y mode l ,

the F i t z h u g h - N a g u m o mode l and of m a n y in-

t egra te -and- f i re mode l s , for e x a m p l e , see [2],

[25] and [13]. In its f avor , the mode l has the

qua l i ty of r e f r a c t o r y n e s s : A f t e r a neu ron fires,

the s t r eng th of exc i t a t ion n e e d e d to ret i re it

d e c r e a s e s wi th t ime.

More usefu l for our p u r p o s e s than the Poin-

car6 map is the L i s s a j o u s (phase p lane) g raph of

the t r a j e c t o r y of the sys t em. The i n s t a n t a n e o u s

s ta te of o sc i l l a to r Y is p lo t t ed aga ins t that of

osc i l l a to r X. Fig. 3 dep ic t s a typ ica l case .

A s s u m e wi thou t loss of gene ra l i t y : (1) The

b a s e l i n e - t o - t h r e s h o l d d i s t ances , the s lopes , and

the coup l ing f ac to r s a re sca led into a uni t

square . (2) P h a s e is m e a s u r e d l inear ly f rom ze ro

to un i ty a long each axis of the unit square . (3)

The t r a j e c t o r y s ta r t s at the lower left co rne r of

the unit square , the "o r ig in" , w h e r e bo th X and

Y are at the base l ine level s imu l t aneous ly . Wi th

regard to (3), we de fe r c o n s i d e r a t i o n of t r ans i en t

cyc le s to sec t ion 5.

Y~

>-

Qyx~

/ -1L

X2 xl X4 I X Phose

Fig. 3. The lissajous figure corresponding to a frequency ratio of Wr/W x = 10/7. The retrace states generate series x; and Yk. Coupling magnitudes axy and ayx are measured rela- tive to the upper, right corner of the unit square.

308 T. A l l e n / C o u p l e d neurons as a latt ice on R 2

When a retrace state of one of the oscillators occurs, the concurrent state of the other oscil- lator can be read directly from the lissajous figure. For example, in fig. 3, when Y first strikes its threshold (y = 1) at time l/Wy, X is at phase Xl. Subsequently, when X first strikes its threshold (x = 1) at time 1/w~, Y is at phase y~. Ignoring for the moment possible effects of coupling, the system proceeds to generate

sequences x i and Yk:

x i = j w y / w , ( m o d l ) , j = 0 , 1 , 2 , 3 . . . . . ( la)

and

Yk - kw~/wy(mod 1), k = 0, 1,2, 3 . . . . ( lb)

and the cycle closes if and only if Wy/Wx is rational, i.e., iff x i = Yk = 1 = 0 for some j and k. {Note: ( la) satisfies xi~ - xi + % / w x (mod 1)}.

Now consider the effect of coupling. At each

event xj and Yk, a pulse of magnitude axy into Y or magnitude ayx into X respect ively occurs. These magnitudes are conveniently measured, as shown in fig. 3, as orthogonal extensions

from the top right corner of the unit square. The question is: When does one of these extensions first intercept the t ra jectory? That is, on the common elapsed time scale, which xj or Yk first satisfies one of the inequalities

I -- Xj ~ ayx, ( 2a )

o r

1 - Yk <~ axy? (2b)

The instant that one of these conditions is met, the t ra jectory retraces to the origin, and the

cycle recommences . Now for the crucial step. Let the unit square

define a lattice by unit translations in the bipositive quadrant. See fig. 4. The dimensions of the lattice are periods of X and periods of Y - time and t i m e - elapsed f rom the origin. The

4 -I -t ~ Y 3 /

~3//ix3 i 3

/ x2 / Y2

Y

......

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

X Phase

Fig. 4. The lattice corresponding to a f requency ratio of wy/w~ = 10/7. The series xj and Yk here are identical to the series in fig. 3. Coupling magni tudes a~ and a~x are measured relative to the lattice points.

t ra jectory in the lattice is a straight line, a ray having slope Wy/W× from the origin. The suc-

cessive distances xj and Yk from the trajectory to the lattice points are precisely the sequences given by eqs. (la) and (lb). Thus, the algebraic basis of an isomorphism between the phase - plane graph and the lattice is that the con- gruences apply identically to both geometrical viewpoints.

The coupling criterion at each lattice point is identical to the criterion at the upper right cor- ner of the lissajous figure: We must examine the

value of 1 - x i and 1-Yk relative to ayx and axy respect ively in accordance with eqs. (2a) and (2b). Hence we can represent axy and ayx as orthogonal, l ine-segment extensions of points throughout the lattice, congruent to those on the upper right corner of the original unit square. The first instant that the ray intersects one of the extensions, the criterion is fulfilled which resets the system to the origin, and the cycle recommences , exhibiting a ratio given by the coordinates of the terminal, extended lattice

point. The overall picture is this: For each com-

bination of the coupling parameters we can draw a lattice, and plot every ray in the biposi-

T. A l l e n / C o u p l e d n e u r o n s as a la t t i ce on R ~ 309

tive quadrant to its termination on an extended lattice point. Fig. 5 exemplifies unidirectional coupling of X into Y (axy ~ 0.4, ayx ~ 0). Fig. 6 illustrates bidirectional coupling (a~y ~ 0.3, a~ 0.2). Remote points in the lattice are typically "shadowed" completely or partly by extended points nearer the origin. We say that these nearer points are "visible" from the origin. As couplings become smaller than shown, more of the distal ratios, those having large numerators and denominators, become visible. In contrast, as couplings become larger, the simple ratios like 1 : 1 become ever more predominant.

Observe in fig. 6 that the solid angle for each ratio under bidirectional coupling is typically divided into two parts. When the ray terminates on the westward extension, we say that Y "dominates" and X is "subordinate". That is, Y keeps its natural f requency while the final period of X is cut short to meet the common ratio. Conversely, when the ray terminates on a southward extension, we say that X dominates and Y is subordinate.

In summary, the problem of f requency lock- ing reduces to that of expressing constraints on visibility in the extended lattice. Recall, the problem is to determine the map of f requency locking for the interaction, i.e., the region in the

Fly

5

4

3

2

1 2 3 4 5 6 7 N x

Fig. 5. The lattice for unidirectional coupling when axy = 2/5. All rays in the bipositive quadrant are plotted to their terminus on an extended lattice point.

three dimensional parameter space, {w y / w ~ , a ~ ,

ayx}, within which each arbitrary ratio, ny/nx , is obtained.

3. The visibility (phase-locking) map

Now the question arises, if it is possible to systematize the analysis in such a way as to reduce to a minimum the number of equations necessary to express the constraints on visi- bility.

Systematic analysis of such a problem depends on understanding the way in which shadows are cast by one figure on another in the lattice. Empirical study, say, of diagrams 5 and 6, reveals that a consistent algebraic relation holds between the integer-valued coordinates (a, b) of the figure that casts the critical shadow and the integer-valued coordinates (p, q) of the figure on which that shadow is cast. This is always the unimodular relation, a q bp = +-1.

The analysis in [17] bears out this empirical observation. The positions of the constraints on visibility are specified by this unimodular rela- tion regardless of the size or shape of the figure, within certain weak restrictions. The figure we have here satisfies these. The restrictions are: (1) The figure located at point (p, q) must con- tain the point (p, q); (2) it must be connected, i.e., it can be drawn without lifting pencil from paper; and (3) it must be contained within a square, the corners of which are the four points nearest to (p, q).

To see where the unimodular relation comes from, consider the distance from an arbitrary grid-point (a, b) to an arbitrary ray of slope n y / n x = p / q . Assume that p and q are coprime. The vertical distances are

dy = a - b ( p [ q ) = ( a q - b p ) / q (3a)

and the horizontal distances are

dx = b - a ( q / p ) = ( b p - a q ) / p . (3b)

31(1 T. A l l e n / C o u p l e d n e u r o n s as a l a t t i c e on R 2

6' 7 7 7 7 7 7

5 J~ ~ I 7 7 7 7

ny 3 7 7 7 7

2 7 7 7 7

1 , ~ ~ = 7 7

1 2 3 4 5 6 7 nx

etc.

Fig. 6. The lattice as it appears for bidirectional coupling when a~s. 3/10 and a> = 1/5.

disjoint sets of slopes; and (3) only figures at coprime coordinates can ever be visible, because of the shadowing effect of figures col- inearly nearer the origin. We slate our main result as

Theorem 1. If n,. and nx are coprime and >0, then wylu'~C V(ny, n , a~, ax,.) if, and only if

(n~ a~O/n~-< w,/w~ ~ ny/(n~- ayO (4a)

and

(n~ a~y)/n" ~ Wy/W~ :~ n"/(n"- a>), (4b)

The dx are positive for points above the ray, and the dx are positive for points to the right of the ray. The Diophantine equations in the numerators determine two unique points that

are closer to the origin than (p, q): One point that minimizes dy > 0, and one that minimizes dx > 0. Denote these points by (n'y, n'0 and (n',', n"). These are the solutions of a q - b p - + l and of bp - aq = + 1, respectively. As shown in [17], these points are the closest, not only to the ray of slope p/q, but also to every ray having its slope in the radial set, n'~/n',> wy/w~>n','/n~. Therefore , the extensions of these two points are the only ones we need to consider with respect to shadows on the extensions of (ny, n×).

In the present instance, we denote by V ( n y ,

n×, tly×, tl×y) the set of slopes that comprise the "vis ible" set for a figure located at (ny, nO. A slope, Wy/W, is an element of the set if, and only if the ray at that slope from the origin intercepts the figure located at at (ny, nx) before intercepting any other figure in the lattice. This notion of visibility is both intuitive and in keeping with the notion of visibility in point- lattices [18]. The notation emphasizes that the visible set depends on the parameters of coup- ling. It is intuitive (1) that the visible set is a subset of the radial set of the previous paragraph; (2) that the visible set is connected, that is, no figure appears from the origin as two

where n~, n'x, and n~', n" are the solutions of

n~nx- n~ny- +1 (4c)

and

n~nx- n~n~- - I (4d)

in integers such that n~ < ny, n', < n , n ' ~ n~ and c t J

n x < n x.

The closed constraints, eq. (4a), say that a ray in the set must intercept the extensions of the p o i n t (ny, n×). It must occur as an intercept on or below the tip of the westward extension of (ny,

nx), and it must intercept on or above the tip of the southward extension of (ny, nO- The open constraints, eq. (4b), say that no ray in the set can intercept the extensions of the points (n~, n'0 or (n~', n"). That is, the slope must be strictly below the slope subtended by the tip of the southward extension of (n~, n'x), and it must be strictly above the slope subtended by the tip of the westward extension of (ny, n").

The coordinates of the points (n¢, n'0 and (ny, n") and (ny, nx) satisfy a suite of algebraic relations that are customarily introduced as propert ies of Farey series [18]. These propert ies will emerge in the section to follow, where we show how to draw a map of the zones of phase

T. A l l e n ~ C o u p l e d neurons as a lat t ice on R 2 311

locking for this model and examine the structure

of the map.

4. Structure of the visibility (phase-locking) map

Theorem 1 states the domain of parameter values on R 3 over which any given ratio is

obtained. In fig. 7, exploded isometric represen- tations of V(1, 1, axy, ayx), V(I, 2, axy, ayx) and V(2, 3, axy, ayx) are drawn to convey an idea of what these regions look like in R 3 and the ele- gant manner in which they fit together. Fig. 8 shows several zones on R 2 for the special case

Wy :~

1/ "',

Fig. 7. Exploded perspective view of the three-dimensional zones W/l / I ] , W/l/2], and W[2/3]. W[I/1] is shown shifted down and to the right, while W[2/3] is shifted up and to the left.

of ayx = 0 , 0 4 axy<~ 1, that is, unidirectional coupling of X into Y. Fig. 9 shows several zones on R 2 for the special case of axy = ayx = a, 0

a ~< 1, that is, isolateral coupling between X and Y. Of course, only a few of the simplest, largest zones are shown. The structure ramifies without limit as the coupling factors approach zero.

Now, suppose one is interested in delineating the zone of phase locking for some particular ratio, n y / n x , a s a functions of the parameters . For example, what is V(10, 7, axy = ayx = a ) ?

The following algorithm uses the continued fraction algorithm to solve the Diophantine eqs. (4c) and (4d), and eqs. (4a) and (4b) to plot the boundaries of the zones. Ideas for imple-

2A

7/S 4/3

. ~ 1/I

4/5 3/4

7 3.. 3..3~ r,/9 -3~

gN 1/4 1/5

~ 2:1

0 axy

Fig. 8. Some low order zones on the phase-locking map for unidirectional pumping of Y by X, when ayx = 0, and 0 < axy "~ I.

o 1 a

Fig. 9. Some low order zones on the phase-locking map for isolateral coupling when axy = ayx = a, and 0 < a < 1.

menting "seminumerical a lgori thms" can be found in [19]. See Olds [26] for background.

312 T. Allen/Coupled neurons as a lattice on R 2

Algorithm A (l) Use Eucl id ' s a lgori thm to de termine the

coefficients of the con t inued f rac t ion for p/q = 10/7, and to ver i fy that (p, q) = (10, 7) = I.

1 0 = 7 × 1 + 3

7 = 3 × 2 + 1

3 = 1 × 3 / /

- - - > 10/7 = [1, 2, 3].

(2) Calcula te the cumulants of the cont inued

fract ion using the recurs ion pg = a~p~ t + P~ 2 and

qi = aiqi i + qi 2. Let P t = 1, P 2 = 0, q ~ = 0, q 2 = 1 to get the recurs ion started.

f UO, d I, . . . . . . . , am ] P/q = tPo/qo P~/ql P m / q m '

2 3] 1 0 / 7 = 1 3/2 10/7 "

The next- to- the- las t cumulant , Pm J/qm ~, is equal to n'y/n'x if m is even, and it is equal to

t t t t r i ny/nx if m is odd. Here we find ny/nx = 3/2.

(3) Calculate ( P i - P i O/(qi- qi-l). This equals n;/n" if m is even, n;/n" if m is odd. Here

n;/n; = 7/5. (4) The in ter ference on V(10, 7, a) due "to the

sou thward ex tens ion of (3, 2) begins when the

first const ra in t of eq. (4a) intersects the first

const ra in t of eq. (4b), that is, when ( 3 - a)/2 = 10 / (7 - a). Solving for a, we find a = 5 - 2~/6

0.1010. Plot the line Wy/W~ = ( 1 0 - a)/7 f rom a =

0 to a ~0 .1010. (5) Similarly, the in ter ference on V(10, 7, a)

f rom the wes tward extens ion of (7, 5) begins when the second contra in t of eq. (4a) intersects the second cons t ra in t of eq. (4b), that is, when 7/(5 - a) = (10 - a)/7. Solving for a, we find a = ( 1 5 - ~ / 2 2 1 ) / 2 ~ 0 . 0 6 7 0 . Plot the line, Wy/W~= 1 0 / ( 7 - a) f rom a = 0 to a =0.0670.

(6) The in ter ference f rom two sides reduces V(10, 7, a) to a null set exact ly where the two const ra in ts of eq. 4b intersect , that is,

where ( 3 - a ) / 2 = 7 / ( 5 - a ) . Solve for a to find

a = 4 ~ / 1 5 ~ 0 . 1 2 7 0 . Plot the line w~/wx

( 3 - a ) / 2 f rom a ~0 .1010 to a = 0.1270. Plot the line wy/w~= 7/ (5- a) f rom a ~ 0 . 0 6 7 0 to a ~ 0.1270.

Referr ing to fig. 8b, obse rve that the four lines drawn in this way fully delineate V(10, 7). The

critical points are there. Besides its usefulness as an aid in solving the

linear Diophant ine equat ion, the cont inued

fract ion informs us about the s t ructure of visi-

bility in lattices. Suppose that the ratio of natural f requencies

is held cons tan t as the parameters of coupling

increase f rom zero. Which ratios does sys tem at tain? This ques t ion is equivalent to examining

the success ion of zones on figs. 7 or 8 or 9 that

are in tercepted by a horizontal plane or line, wy/wx = constant . The ratios that occur will al-

ways co r re spond to convergen t s of the con- tinued fract ion, and to no other ratios. Not all of

the convergen t s need occur , and this depends on the relative magni tudes of the coupl ing fac-

tors. For example, if w~/wx~ (~ /5-1) /2 , the

golden section, and if the coupl ing is uni- direct ional with ayx=0, then the Fibonacci

ratios 1/1, 2/3, 5/8, 13/21, . . will occur as axy decreases f rom one down to zero. These are the

convergen t s that satisfy Pi/qi > (~/5 1)/2. Addit ionally, the cont inued f rac t ion specifies

the posi t ion of V(p, q) on the visibility map. For

example , to get to the zone cor respond ing to the fifth convergen t of pi, 104348/33215 = [3, 7, 15, 1, 292], one counts out seven zones along the

upper bounda ry of V(3, 1) to reach V(22, 7). Then one counts out 15 zones along the lower

b o u n d a r y of V(22, 7) to reach V(333, 106), and then out one zone along the upper boundary of V(333, 106) to reach V(355, 113), and then 292 zones out along the lower bounda ry of V(355, 113) to reach V(104348, 33215). Exercise : Apply this interpretat ion to V(10, 7) and V(5, 8).

Algor i thm A is inefficient if we desire to plot a

large number of ad jacent zones. There is no need to solve the Diophant ine equat ion

repeatedly.

T. A l l e n / C o u p l e d neurons as a lat t ice on R 2 313

Algorithm B (1) Start with two fractions that are uni-

modular, such as c/d = 0/l and a/b = l/0 (or say, arbitrarily, a/b = 333/106 and c/d = 355/113). Plot the zones for these ratios, using algorithm A.

(2) Calculate the mediant (a + c)/(b + d) and put this new ratio in between the original ratios, for example, 0/1, l / l , l/0. The new ratio becomes ny/nx and the ratios next to it are automatically the primitives of theorem I. If we are plotting the zones by hand, we only need to plot the primary constraints from eq. (4a), using the n e w n y / n x until they hit the lines for a/b and c/d, which we have already drawn in step 1. If a computer is doing the drawing, it will have to remember the sequence c/d, p/q, a/b and use the appropriate values as in steps 4 and 5 of algorithm A to determine when to stop plotting eq. (4a).

(3) Calculate mediants again from adjacent terms of the previous sequence to yield, for example, 0/1, 1/2, 1/1, 2/1, 1/0. The neighbors of each new ratio in the sequence are always its

t l unique primitives, (n¢, n'x) and (ny, n"), of theorem I. We only need to plot eq. (4a), because the primitives have already been plot- ted. We can see where to stop plotting, or the computer can calculate where to stop as in algorithm A.

(4) Continue in this manner to generate mediants and plot the new zones in relation to their preexisting neighbors.

The above algorithm could proceed in quite a number of different ways, corresponding to a Farey series, a complete series of mediants, or any such series extended by the mediant opera- tion on adjacent ratios [18, 19].

As is true with the continued fractions, the Farey series inform us of the global structure of visibility in the latticework. The branching structure implied by the binary nature of the mediant operator is sometimes termed the Stern-Pierce tree (see [19], p. 363). The tree leading to the Farey series of order 12 is diagramed in fig. 10 to convey graphically its beautiful symmetry, which is not so apparent in the algebraic form. Each node on the diagram

0 1

i i/! / / il/il 1 5 7 11

12 ~ 12 1-2

Fig. 10. Topology of the Farey series of order 12. This structure is the geometric dual of figs. 7 thru 9.

314 T. A l l e n / C o u p l e d neurons as a lat t ice on R 2

(imagine it extended to infinity) is connected by lines to all other nodes with which it has a unimodular relationship. Each node is the mediant of two nodes above it, and each node contributes to the mediant of an infinite number of nodes below it. In terms of phase locking, transitions from a ratio ny/nx to a ratio p / q are possible if, and only if, p /q = ( k n y + ny)/

t t t ! (knx + n '0 or p / q = (kny + n y)/(knx + n ×), where k = 0, 1, 2 . . . . . . because these are the only solutions of n ~ p - n y q = + 1 and n x p -

n y q = - 1, respectively. Note that the structure of connected nodes on fig. 10 is the geometr ic dual of the contiguous zones of figs. 7 thru 9.

The branching structure of the S tern-Pierce tree, via the mediant operator , also clarifies one sense in which the transitions are literally sub- harmonic bifurcations. Although the number of fixed points seldom doubles at a transition be- tween two ratios, the transition always involves moving up or down on the binary structure of the tree.

5. Uniqueness and stability of rational phase locking

From arbitrary initial conditions, do the cou- pled oscillators arrive at the unique cycles im- plied by theorem 1 ?

It is easy to see that the problem of arbitrary initial conditions amounts in the lattice to con- sidering rays that originate at arbitrary points within the unit square closest to the origin. If and when such a ray strikes an extended lattice point, it, or rather the t ra jectory of the system, resets to the origin. If this occurs, we can say that the fixed point cycle is asymptot ical ly unique for the initial condition in question. We must ask if this occurs for all initial conditions within the unit square nearest to the origin. The following argument is along the line of Kronecker ' s theorem in one dimension (c.f. [18], p. 375). In a paragraph to follow we shall relate

these ideas to the Poincar6 map, because the Poincar6 map is the usual approach to questions

of stability in systems of this type. The equation of an arbitrary ray in the x - y

lattice is written, y - u = ( w y / w O ( x - v ) , where (u, v) is the starting point of the ray, with

0 < ~ u < l and 0 ~ < v < l . The vertical distances f rom arbitrary lattice points (j, k) to the ray are

dy = j - u - {(Wy/Wx)(k - v)}. (5)

The distances are positive for points above the ray and negative for points below. A similar formula can be written for the horizontal dis- tances, d~, from the points to the ray. (Eq. (5) reduces to eq. (3a) when wy/Wx = p / q and u = v = 0.)

If wylwx is irrational, then dy takes on arbi- trarily small values over (j, k). This is Kronecker ' s theorem. In fact, the distances

(mod 1) are densely and uniformly distributed over the interval [0, 1), when Wy/Wx is irrational. Afo r t io r i , for every (u, v) and every irrational wy/wx, there exists a (j, k) such that dy < axy (or such that dx < ayx), SO long as axy (or ayx) is greater than zero. This means that the ray eventually hits an extended lattice point. Subsequently, the unique fixed point cycle is attained.

However , if Wy/Wx= p/q , a rational number,

Kronecker ' s theorem does not apply. The ray does not have to come arbitrarily close to lattice points. To see this, rewrite eq. (5) as

d r = {(jq - p k ) / q } - { (uq - pv ) / q } .

Due to the periodicity of the lattice, we may consider the positive fractional parts of the terms. The term e = ( u q - p v ) / q (mod 1) is a real number in [0, 1). The term i/q = ( j q - p k ) / q

(mod 1) takes on the rational values from the set, 0, 1/ q, 2/ q . . . . . ( q - 1)/ q, as the coordinates j and k are varied. Consequently, dy takes on the values

d y E { ( i [ q ) - e}(mod 1), i = 0, 1 . . . . . q - 1. (6)

T. Allen/Coupled neurons as a lattice on R= 315

One of these values is the unique minimum value. Certainly, the minimum value is less than l/q, that is, dy< l/q, for some (L k) in the lattice. Now let us state:

Theorem 2. If Wy/Wx ~ V(ny, nx, axy, ayx) and Wy/Wx ~ ny/nx, then, for all (u, v) there exists some (j, k) for which dx ~< ayx or dy ~< axy.

Proof. The case of Wy/Wx irrational follows directly f rom Kronecke r ' s theorem. We have to prove the proposi t ion for W y / W x = p / q , a

rational number. Assuming the contrary, we have that, for all (j, k),

dy = { ( / ] q ) - e } < axy

> ( n y q - nxp)/q

> l/q,

a contradiction. (The second and third in- equalities follow from Wy/WxC V(.) and wy/wx < nylnx.) Symmetr ical reasoning applies in relation to dx < ayx. []

This leaves one problematical case, that of Wy/W x = ny/n x when W/Wx ~ V(ny, n0. If this is the case, a kind of neutral stability can occur. In the lattice this neutral stability corresponds to a line parallel to the ray of slope ny/n~ f rom the

origin. Recall that there are "cor r idors" of points, defined by eqs. (3a) and (3b), parallel to the line of slope ny/nx in the lattice.

On the visibility maps, figs. 7 to 9, the lines or planes wy/Wx= ny/nx within V(ny, nO are the loci of neutral stability. All other points within V(ny, nO attain the unique fixed point cycle, which includes a s imultaneous retrace of both oscillators: retrace to the origin in accord with theorem 1.

The rate at which a unique fixed point cycle is approached depends on how long a ray has to go before hitting an extension.

Under unidirectional coupling, fig. 8, the lines Wy/W, within V(ny, nO are special. Just above

every V(ny , nx) is an infinite sequence of zones t ! of the series V(kny+ny, knx+nO, for k =

0, 1, 2 . . . . . which for high values of k become progressively narrower and crowded ever closer to the line Wy[Wx = ny/nx. Also, at the points Wy/W~ = ny/nx, a = llnx on the visibility map, all of the zones in such sequences converge to points. For example, examine the point a -- 1/2, wy/Wx = 1[2 on fig. 8. Since each of the zones in the sequence has a different period length, a small amount of noise in the parameters of a system biased near one of these singularities could be amplified into a large uncertainty in

period length. The above conclusions all have counterpar ts

in terms of the Poincar6 mapping. It is easy to see that the circle map for this system, 0i ~ 0i+l, consists of horizontal and diagonal segments only. This is true of any iterate of the mapping. Theorem 2 asserts that under the conditions of the theorem some iterate of the mapping will have fixed points, and that none of the diagonal segments will overlie the identity. If wy/Wx V(ny, nO and wy/wx ~ nflnx, then nx fixed points appear in the nyth iterate of the map X relative to Y, and ny fixed points appear in the nxth iterate of the map of Y relative to X. The cases that exhibit neutral stability will have a diagonal element overlying the identity in some iterate of the map. The cases that are near to being neu- trally stable will have a diagonal segment not

directly on, but arbitrarily close to the identity, and a t rajectory can be caught in the corridor thus formed for an arbitrarily long time. An example is shown in fig. 11.

1

O0 Oi

Fig. I1. A circle-map showing a seven point cycle and illustrating a "corridor" effect. As the corridor approaches the diagonal, the length of the cycle increases without limit.

316 T. Al len /Coupled neurons as a lattice on R 2

6. Discussion

Let us consider the ordering of ratios in table I. Many of the transitions between ratios there are unimodular. For example, in column 1, 28 out of 31 transitions are unimodular. Our model, as well as that of Keener et al. [3, 11, 20] and of

Sato et al. [14, 15, 16], explain this uni- modulari ty, but in different ways. To explore this issue, we compare and contrast the lattice con- struction we have developed here to the Cantor set construct ions of the alternatives. We should mention at the outset that other, non-uni- modular types of ordering evidently occur in the data in table I, for example, there are

sequences of modulari ty = 2 and of modulari ty = 0 (period doubling). These latter effects are best explained by models (see [21], [22] and [12]) in which the systems are represented as non-monotone maps on the torus. In contrast , it seems that unimodular ordering is associated with monotonic maps on the torus. Our contention is that sys tems exhi- biting unimodular ordering have a lattice-like character , of which the model developed here is prototypical .

In surveying empirical data such as that of table I, we have to question whether the

sequences of ratios are complete as they stand, or whether intervening ratios exist that were bypassed in the experiments . For example, concerning the results listed in column 2a of table I, the investigators [6] said, "These and other values not presented here suggest further that [rotation number] is a continuous, mono- tone function whose derivative vanishes almost eve rywhere . " Their inference is supported by theoretical insights from a class of continuous, invertible mappings on the torus [23, 3] and from a class of piecewise-continuous, increasing mappings on the torus [20]. See also [21]. In contrast , the present theory predicts that zones for different, finite ratios can be directly con- tiguous, and fur thermore , it predicts that zones for two ratios a : b and c : d will be contiguous

only if the ratios satisfy the unimodular relation, a d b c = + 1.

On one hand, under the present naive theory, we would predict that the list is nearly complete

as it stands, except that a further experiment would disclose a few, or perhaps many, ad- ditional ratios that would leave the list com- plete, finite, and unimodular throughout. (This assumes absolute stability in the system and infinite resolution in the instruments of measurement , but we shall not address this problem here.) For example, the simplest out- come of the experiment in accord with the hypothesis would disclose that the sequence

-3 / i 0 - -5 /16 - - i l3 - -215- -419- -6 /13 -

reported in table I, column 1, should really be

- 3 / 1 0 - - 4 / 1 3 - - 5 / 1 6 - - 1 / 3 - - 2 / 5 - - 3 / 7 - - 4 / 9 - - 5 / 1 1 - - 6/13-.

There could be other outcomes in accord with the hypothesis , but these would always involve a finite number of ratios. The possibilities, when they are not obvious, can be found with the aid of simple continued fractions. For example, 3/10 = [0, 3, 3] and 3/16 = [0, 3, 5], so at least one intervening ratio must be [0, 3, 4] = 4/13. Or, for another example, take the transition from 5:17 to 3:11 in column two of table I. We have 5 / 1 7 - [0, 3, 2, 2] and 3/11 = [0, 3, I, 2], so [0, 3, 1, 1] = [0, 3, 2] = 2/7 is one missing ratio. In- cidentally, ratios could be rejected too, e.g., take away 5/16 from - 3 [ 1 0 - - 5 / 1 6 - - 1 / 3 - . )

On the other hand, under the somewhat more realistic models by Sato et al. [14, 15, 16] and by Keener et al. [3, 11, 20], further exper iment would detect additional ratios at every position in the list. Ultimately, given absolute resolution of the system and the instruments, the at tempt to complete the list would become an at tempt to list the real numbers as fractions. Why, then, are there so few ratios in the empirical lists, and why are so many of the transitions unimodular?

T. A l l e n / C o u p l e d neurons as a latt ice on R 2 317

Table I Data f rom diverse sources , showing ratios obtained as funct ions of certain parameters . If two ratios of phase locking are a[b and c/d, then we define their modulari ty as M = ad - bc. The modulari ty of neighboring ratios in each sequence are listed in the offset columns. The co lumns are: (1) Ha rmon [5] studied an electronic analogue of neurons . Excitory pulses from one device " s y n a p s e d " to another device. The ratios were obtained as the s trength of coupling was varied. (2) Flaher ty and Hoppens teadt [6] simulated the van der Pal equation with a sinusoidal forcing term on a digital computer . The ratios were obtained as the tuning parameter was varied. The amplitude of forcing was greater for the sequence of column 2b than for that of column 2a. (3) Gollub et al. [7] coupled two oscillators made with tunnel diodes. The ratios were obtained for an excitory mode of coupling as the natural f requency of one oscillator was varied. See Hoppens teadt [8] for an analysis of this circuit in relation to the F i t zhugh -Nagumo model of neurons. (4) Perker et al. [9] st imulated a stretch receptor of a crayfish with a train of inhibitory post°synaptic potentials (IPSP). The ratios were obtained as the f requency of the pulses was varied. (5) Guevarra et al. [10] st imulated aggregates of embryonic heart cells in culture with trains of pulses of depolarizing current. The ratios were obtained as the f requency of st imulation was varied. Their s tudy focused on the period-doubling sequences

(1) (2a) (2b) (3) Electronic Van der Pal Van der Pal Tunnel neuron equation equation diodes

1:1 4:5 3:4 5:7 2:3 3:5 1:2 6:13 4 :9 2:5 1:3 5:16 3:10 2:7 3:11 1:4 3:13 2:9 3:14 1:5 3:16 2:11 3:17 1:6 3:19 2:13 1:7 2:15 1:8 2:17 1:9 2:19 1:10

1:3 1 7 :8 3:10 1 13:15 5:17 4 6 :7 3:11 2 11:13 5:19 I 5 :6 1:4 1 9:11 3:13 1 4:5 2 :9 1 7 :9 1:5 2 3:4 1:7 2 5:7 1:9 2 2:3 1:11 2 3:5 1:13 2 1:2 1:15 2 1:3 1:17 2 1:5 1:19 2 1:7 1:21 2 1:9 1:23

(4) Neural pacemaker

2:1 1 3:2 1 1:1 1 1:2 1 1:3

3:2 7:5 4:3

13:10 9:7

14:11 5:4

11:9 6:5

(5) Heart pacemaker

2/1

(m + l ) / m series

1:1

period doubling

extra beats

2/3

318 T. A l l e n / C o u p l e d n e u r o n s as a lat t ice on R 2

We must digress to explain that unimodulari ty also plays an important role in these latter theories. The parameter interval on R ~ is frag- mented in the manner of a Cantor set. At every stage in the construct ion of the Cantor set there exist subintervals of the original interval having

ratios assigned to them. Any two of these subintervals (say for ratios a/b and c/d) are separated by an interval as yet having no ratio assigned to it. A new interval is constructed in the interior ("middle 1/3) of each unassigned interval and is assigned the "med ian t " ratio, (a + c) /(b + d). Observe that if ad - bc = 1, then ( a + c ) d - ( b + d ) c = ( b + d ) a = ( a + c ) b = l, so the new ratio is unimodular with its neighbors.

The construct ion proceeds in this manner ad infinitum along all its branches. Because the original ratios are 0/1 and 1/1, by induction the construct ion is arithmetically equivalent to a Farey- type series, the binary branching struc- ture of which, via the mediant operator , is the S tern-Pierce tree we have examined in section 4. Sato [15] has developed this arithmetical per- spective explicitly in his "node of period m " construction. In the other formulat ions it is more implicit.

I

In the present formulation, the Farey series also plays an important role, but the construc- tion does not involve a Cantor set. It involves visibility in a lattice. Moreover , here in effect the Farey- type series usually terminates. What this means is that zones for finite ratios are typically directly contiguous, and therefore that the ratios of phase locking can take discrete jumps as the parameters vary. In the other formulat ions, the series bifurcates incessantly along all its branches, leading to irrational- valued limits. This implies that the ratios of phase locking (or the rotation numbers) are continuous functions of the parameters , whose derivative vanishes almost everywhere , as was stated in [6] in relation to the data in column 2a and 2b here. The construct ion of the Cantor set can reduce the irrationals either to a set of measure zero or to a set of finite measure on the

parameter interval (see [2]). But the main arithmetical distinction between the present model and the others mentioned above concerns the termination or nonterminat ion of the Stern- Pierce tree, and does not concern unimodular

ordering per se. The solution here is similar enough to a Can-

tor set to have heuristic value in relation to the other formulations. The essentially unimodular ordering found here, represented graphically, for example, in fig. 8b, would not necessari ly be altered if irrational valued lines or regions were to intercalcate away from the axis, where coup- ling equals zero, into or across the structure of rational-valued zone. (Note that by theorem 2 irrational-valued cycles are absolutely dis- allowed in our model when the coupling is non-zero.) Formally, it seems that uni- modular i ty-preserving t ransformations could bring a whole class of maps of phase locking into correspondence. For example, the maps of phase locking given in [2], fig. 5 and further implied by the analysis there could evidently be

thus t ransformed. However , such t ransformat ions might entail

drastic changes in the shapes of the zones, even while conserving the unimodulari ty of con- tiguous zones. That is to say, the unimodular ordering specified by eqs. (4c) and (4d) may be more robust than the shapes of zones specified by our eqs. (4a) and (4b). As another example of this, observe that although the neighboring ratios in column 1 of our table | are largely unimodular as a function of "exci tat ion strength", they are not successive cumulants of a continued fraction, as they would be under the present model, as explained in section 4.

Now, returning f rom the digression, we ask why there are so few ratios in the empirical lists, and why so many of the transitions are unimodular. It is possible to imagine otherwise on both counts. The answers probably lie in the prediction that zones for simple ratios like 2:3 are generally bigger than zones for "distal" (in terms of a lattice) ratios like 131 : 147. In fact, all

T. A l l e n / C o u p l e d n e u r o n s as a la t t i ce on R 2 319

of the theories agree in this. If this is so, in order to report a non-unimodular pair in a list, an experimentor would have to miss a relatively

large zone and hit instead the smaller zone

corresponding to a more distal ratio. For exam-

ple, in column 1, the zone for the ratio 4/13 was

presumed missing between 3110 and 5/16. Under

the models, the zone for 5/16 should be smaller than the zone for 4/13. However , the real-world

system might in fact be perverse. Indeed, in the

data of column 2a, the zone for 1/5 was the

widest, wider than the zone for 1/4. But the trend for zones to be smaller distally is

generally true, especially when the sizes of neighboring zones of are compared locally.

In columns 2a and 2b of table I, where ad-

ditional, unlisted ratios are know to occur, the

evident unimodularity could result simply from

the investigators' decision to report the largest

zones in each local interval of parameter varia-

tion. Similarly, in the data from the biological pre-

parations in columns 4 and 5, the zones were

separated by zones that were not identified with any particular ratio. This effect may be due to

noise [24, 25], which may smear out the narrow

zones corresponding to the distal ratios. In conclusion, we contrast the Sarkovskii

ordering with the unimodular kind. The period

doubling in the data in column 5 of table I (data from [10]) has been reproduced in a tractable

model by Guevarra and Glass [12]. Fig. 10 of [12] shows a diagram having the same meaning

as our fig. 10. Both diagrams specify which ratios (on Z 2) are connected. Both are the

geometric duals of the contiguity of zones on the maps of phase locking. But the arithmetical

structures are very different. This may be explained by noting that the period doubling

arises from a non-monotonic map on the torus, whereas unimodular ordering arises from monotonic maps on the torus.

Moreover, the model in [12] has regimes of parameter values over which its map on the torus is monotonic, and over which its ordering

is presumably unimodular. Thus, one system

can behave very differently over different regimes of parameter values, and can be very

self-consistent within those regimes.

The present results underscore the lattice-like

quality of unimodular ordering where it occurs.

Acknowledgements

The author thanks the anonymous reviewer for many helpful criticisms, especially for those

regarding comparisons with other models of neurons. This work was supported in part by

National Science Foundation grant #DEB-27072

to Dr. George Oster. The author thanks Dr.

Oster for support and guidance. Thanks also to

Art Winfree for help in the formative stage of

these ideas.

References

[1] H. Rademacher, Symposium on Harmonic Analysis & Related Integral Transforms, Final technical report, Cornell University (1956).

[2] J.P. Keener, F.C. Hoppensteadt and J. Rinzel, SIAM J. Appl. Math. 41 (1981) 503.

[3] L.O. Chua, Introduction to Nonlinear Network Theory (McGraw-Hill, New York, 1969).

[4] B. Van der Pol, J. Franklin Inst. 225 (1953) 475. [5] L. Harmon, Kybernetik 1 (1961) 89. [6] J.E. Flaherty and F.C. Hoppensteadt, Studies Appl.

Math. 58 (1978) 5. [7] J.P. Gollub, T.O. Brunner and B.G. Danly, Science 200

(1978) 48. [8] F.C. Hoppensteadt, in Mathematical Aspects of Phy-

siology (Math. Assoc. of America, Washington D.C., 1981).

[9] D. Perkel, J. Schulman, T. Bullock, C. Moore and J. Segundo, Science 145 (1964) 61.

[10] M. Guevarra and L. Glass, Science 214 (1981) 1350. [I1] J. Keener, J. Math. Biol. 12 (1981) 215. [12] M. Guevara and L. Glass, J. Math. Biol. 14 (1982) 1. [13] A. Winfree, The Geometry of Biological Time, Bioma-

thematics, vol. 8. (Springer, New York 1980). [14] J. Nagumo and S. Sato, Kybernetik 10 (1972) 155. [15] S. Sato, Kybernetik 11 (1972) 208. [16] S. Sato, M. Hatta and J. Nagumo, Kybernetik 16 (1974)

1. [17] T.T. Allen, American Math. Monthly, (submitted). [18] G.H. Hardy and E.M. Wright, An Introduction to the

320 T. A l l e n / C o u p l e d neurons as a lattice on R 2

Theory of Numbers, 5th ed. (Clarendon Press, Oxford, 1979).

[19] D. Knuth, The Art of Computer Programming, vol. 2 (Addison-Wesley, New York, 1981).

[20] J. Keener, Trans. Am. Math. Soc. 261 (1980) 589. [21] J. Guckenheimer, Physica 1D (1980) 227. [22] P. Collet and J.P. Eckmann, Iterated Maps of the

Interval as Dynamical Systems (Birkh~user, Basel, 1980).

[23] E. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).

[24] G.P. Moore, D.H. Perkel and J.P. Segundo, Proc. San Diego Symposium for Biomedical Engineering (1963) 184.

[25] L. Glass, C. Graves, G. Petrillo and M. MacKay, J. Theor. Biol. 86 (1980) 455.

[26] C.D. Olds, Continued Fractions (Math. Assoc. of America, Washington D.C., 1963).