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Structures of attractors in periodically forced neural oscillators
Volume 116, number 7
PHYSICS LETTERS A
30 June 1986
STRUCTURES OF ATTRACTORS IN PERIODICALLY FORCED NEURAL OSCILLATORS K. AIHARA a, T. NUMAJIRI a, G. MATSUMOTO b and M. KOTANI a a Department of Electronic Engineering, Faculty of Engineering, Tokyo Denki University, 2-2 Nishiki-cho, Kanda Chiyaoda-ku, Tokyo 101, Japan b Electrotechnical Laboratory, Division of Information Science, Analogue Information Section, Tsukuba Science City, Ibaraki 305, Japan Received 20 March 1986; accepted in revised form 28 April 1986
Periodically forced oscillations in the membrane potential of squid giant axons are analysed by stroboscopic plots at multi-phases of the forcing function. The analysis clarifies the global structures of the attractors in the phase space R2 x S1.
It is well known that periodically forced neural oscillators respond not only periodically but also non-periodically [1-11]. Poincar6 sections of periodically forced oscillations can be easily displayed on a two-dimensional plane R 2 by a stroboscopic plot, or by observing the states of the oscillators only at a fixed phase of the periodic force. Moreover, global structures of the attractors in the phase space R 2 x S1 can be elucidated by displaying stroboscopic plots at multi-phases of the periodic force. Giant axons of the squid Doryteuthis bleekeri were used in this study. A self-sustained oscillation was induced by immersing the axons in a 1: 9 mixture of natural sea water and 550 mM NaC1 [12]. The squid giant axons in this state of self-sustained oscillation behaves as a nonlinear neural oscillator [7,12]. The neural oscillator was periodically forced by a sinusoidal current A sin(2~rFt) applied through an internal current electrode. The amplitude A and the frequency F of the sinusoidal forcing function were changed as the bifurcation parameters. The membrane potential V was recorded through a pair of glass pipette Ag-AgC1 electrodes. The values of the membrane potential V and that of its time differential dV/dt were stroboscopically plotted on the two-dimensional plane V× dV/dt at a phase of the sinusoidal force [8-10]. The stroboscopic plot represents a Poincar6 section of the neural attractor at the
corresponding phase [6-11]. In this report, we display stroboscopic plots at the phases 3 0 ° x k (k = 1, 2 , . . . , 12), to elucidate the global structure of the attractors in the phase space V x dV/dt x S1. It has been shown both experimentally and theoretically that periodically forced oscillations in squid giant axons can be qualitatively classified into (1) synchronized oscillations, (2) quasi-periodic oscillations and (3) chaotic oscillations [3-5,7-10]. When F (the forcing frequency) is close to m/n (a simple rational number) times F N (the naturally oscillating frequency), the neural oscillator usually responds with a n/m synchronized oscillation [7,9,10]. The n/m synchronized oscillation is a periodic oscillation such that n action potentials are generated at locked phases during m cycles of the sinusoidal forcing function. As the fundamental period of the n/m synchronized oscillation just equals m times the period of the forcing function, the stroboscopic plot is composed of m distinct points. Fig. 1 shows stroboscopic plots at multi-phases of a 2 / 3 synchronized oscillation. As each Poincar6 section in fig. 1 is composed of points, the trajectory of the synchronized oscillation is in the form of a limit cycle in the phase space V x dV/dt x S1. Fig. 2 shows stroboscopic plots of a quasi-periodic oscillation. Each stroboscopic plot of the
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
313
Volume 116, number 7
PHYSICS LETTERS A
60 °
30 °
30 June 1986
120 °
90 °
Ikl
150
180"
°
dV/dt
210
14
°
240"
w*t
le 0
270 °
300 °
330"
360"
kO
V Fig. 1. Stroboscopic plots of a 2 / 3 synchronized oscillati#n in squid giant axons (the amplitude A of the sinusoidal force = 4 ~tA, the frequency F of the sinusoidal force = 306 Hz and the naturally oscillating frequency F N = 197 Hz). The number in each stroboscopic plot shows the corresponding phase ( o ) of the sinusoidal force.
30 °
60
©0 °
°
120
°
I
:,"
•
•
'~'~,~ . , . ,
150
180
°
dV/dt
o 1 Ch°
°
'i
.,'?"
.
.
.
, ~v -'~>
240* ::.<. ,.
.d::
!. "1~, .
i
./
/t
=
¢ .i
270 °
330*
300"
360* :.,L
L:;
< . ............i
x-~
~''"
;~
'i
/"
V Fig. 2. Stroboscopic plots of a quasi-periodic oscillation in squid giant axons (A = 0.5 ~A, F = 250 Hz and FN = 203 Hz). 314
Volume 116, number 7
PHYSICS LETTERS A
60"
30"
30 June 1986
90 °
[120" .,.':.
~i~,. ~"
I 50"
dr/or
j#..
180 °
.,,.x...
i
2i0"
,..... '\
240" ¢'.
1.
,,,,;. >. ~..,~
.~,
,~, . f • 4"/
270"
300 ° . r : . , ,-•
330"
360 °
:,
V Fig. 3. Stroboscopic plots of a chaotic oscillation in squid giant axons (A = 2 p,A, F = 266 Hz and F N = 213 Hz).
90"
60"
30"
.r;. y..
120 °
.
, ;-'A
#.,
,~,,,,
1500
1_80"
240"
210" ...,,., -:.:,.
,.j
t270
•
¢.
'
i-
300"
360"
330"
.,',.
:...,•.
.,.
i
.~
V Fig. 4. Stroboscopic plots of a chaotic oscillation in squid giant axons (A ffi 1.5 I~A, F = 270 Hz and F~ ffi 200 Hz).
315
Volume 116, number 7
PHYSICS LETTERS A
merged with another part of the surface (at 150 ° in fig. 4 and at 180 ° in fig. 5) during one period of the forcing function. These processes are similar to those observed in the forced van der Pol oscillator [13,14]. Stroboscopic plots at multi-phases of the forcing function have illustrated that a synchronized oscillation, a quasi-periodic oscillation and a chaotic oscillation in squid giant axons are represented by a limit cycle, a two-dimensional torus and a strange attractor in the phase space V x d V / d t x S 1, respectively. As this method can elucidate the global structures of the attractors, it must be effective for analysing experimental observations of routes to chaotic oscillations such as collapse of tori [7, 10,15-17] in forced systems.
quasi-periodic oscillation asymptotically depicts a closed curve [8,10]. Therefore, the trajectory of the quasi-periodic oscillation is in the form of a twodimensional torus in the space V x d V / d t x S ~. Fig. 3 shows stroboscopic plots of a chaotic oscillation. The stroboscopic plots at the phases 30 ° and 60 ° show that this forced oscillation almost synchronizes to the periodic force at these phases. But the attractor is then stretched (60° 120°), folded (120 ° ~ 2 4 0 °) and compressed (270°--* 360 ° ) during one period of the force. Similar dynamical processes were also reported on the chaotic behavior in the Onchidium giant neuron [6]. The stroboscopic plots at the phases from 150 ° to 240 ° in fig. 3 clearly show that the strange attractor has a structure with multi-layers. Figs. 4 and 5 show another type of chaotic oscillation. While the attractors keep almost tubular structures, a part of the surface seems to be stretched (at 90 ° in fig. 4 and at 120 ° in fig. 5), folded (at 120 ° in fig. 4 and at 150 ° in fig. 5) and
30*
30 June 1986
The authors wish to thank M. Ichikawa for his help in the experiments. A part of this research is financed by the Research Fund of Center for Research at Tokyo Denki University.
90'
60 °
120
°
240
°
36U
°
..'. - .:.
~ H
~7'"
150"
~
...'
,:.,:
i
o¢--~ U-
210
o
°
-.,.. S¢ ~" . a . . ,
dV/dt
•, . ~ ' i . " " -
:270
°
.i
" :'~
. • ~,~*~!~.-"
:300 °
330"
,,,
t
V Fig. 5. Stroboscopic plots of a chaotic oscillation in squid giant axons (A = 2.35 ~A, F = 270 Hz and F N = 1 8 3 Hz).
316
Volume 116, number 7
PHYSICS LETTERS A
References [1] H.R. Hirsch, Nature 208 (1965) 1218. [2] I. Nemoto, S. Miyazaki, M. Saito and T. Utsunomiya, Biophys. J. 15 (1975) 469. [3] A.V. Holden, Biol. Cybern. 21 (1976) 1. [4] R. Guttman, L. Feldman and E. Jakobsson, J. Memb. Biol. 56 (1980) 9. [5] G. Matsumoto, K. Kim, T. Uehara and J. Shimada, J. Phys. Soc. Japan 49 (1980) 906. [6] H. Hayashi, S. Ishizuka, M. Ohta and K. Hirakawa, Phys. Lett. A 88 (1982) 435. [7] K. Aihara, G. Matsumoto and Y. Ikegaya, J. Theor. Biol. 109 (1984) 249. [8] G. Matsumoto, K. Aihara, M. Ichikawa and A. Tasaki, J. Theor. Neurobiol. 3 (1984) 1.
30 June 1986
[9] K. Aihara, G. Matsumoto and M. Ichikawa, Phys. Lett. A 111 (1985) 251. [10] K. Aihara and G. Matsumoto, in: Chaos, ed. A.V. Holden (Manchester Univ. Press, Manchester, 1986). [11] H. Hayashi, S. Ishizuka and K. Hirakawa, preprint. [12] K. Aihara and G. Matsumoto, J. Theor. Biol. 95 (1982) 697. [13] R. Shaw, Z. Naturforsch. 36a (1981) 80. [14] J.M.T. Thompson and H.B. Stewart, Phys. Lett. A 103 (1984) 229. [15] K. Kaneko, in: Chaos and statistical methods, ed. Y. Kuramoto (Springer, Berlin, 1984) p. 83. [16] M. Sano and Y. Sawada, Phys. Lett. A 97 (1983) 73. [17] H.G. Schuster, Deterministic chaos (Physik-Verlag, Weinheim, 1984).
317
PHYSICS LETTERS A
30 June 1986
STRUCTURES OF ATTRACTORS IN PERIODICALLY FORCED NEURAL OSCILLATORS K. AIHARA a, T. NUMAJIRI a, G. MATSUMOTO b and M. KOTANI a a Department of Electronic Engineering, Faculty of Engineering, Tokyo Denki University, 2-2 Nishiki-cho, Kanda Chiyaoda-ku, Tokyo 101, Japan b Electrotechnical Laboratory, Division of Information Science, Analogue Information Section, Tsukuba Science City, Ibaraki 305, Japan Received 20 March 1986; accepted in revised form 28 April 1986
Periodically forced oscillations in the membrane potential of squid giant axons are analysed by stroboscopic plots at multi-phases of the forcing function. The analysis clarifies the global structures of the attractors in the phase space R2 x S1.
It is well known that periodically forced neural oscillators respond not only periodically but also non-periodically [1-11]. Poincar6 sections of periodically forced oscillations can be easily displayed on a two-dimensional plane R 2 by a stroboscopic plot, or by observing the states of the oscillators only at a fixed phase of the periodic force. Moreover, global structures of the attractors in the phase space R 2 x S1 can be elucidated by displaying stroboscopic plots at multi-phases of the periodic force. Giant axons of the squid Doryteuthis bleekeri were used in this study. A self-sustained oscillation was induced by immersing the axons in a 1: 9 mixture of natural sea water and 550 mM NaC1 [12]. The squid giant axons in this state of self-sustained oscillation behaves as a nonlinear neural oscillator [7,12]. The neural oscillator was periodically forced by a sinusoidal current A sin(2~rFt) applied through an internal current electrode. The amplitude A and the frequency F of the sinusoidal forcing function were changed as the bifurcation parameters. The membrane potential V was recorded through a pair of glass pipette Ag-AgC1 electrodes. The values of the membrane potential V and that of its time differential dV/dt were stroboscopically plotted on the two-dimensional plane V× dV/dt at a phase of the sinusoidal force [8-10]. The stroboscopic plot represents a Poincar6 section of the neural attractor at the
corresponding phase [6-11]. In this report, we display stroboscopic plots at the phases 3 0 ° x k (k = 1, 2 , . . . , 12), to elucidate the global structure of the attractors in the phase space V x dV/dt x S1. It has been shown both experimentally and theoretically that periodically forced oscillations in squid giant axons can be qualitatively classified into (1) synchronized oscillations, (2) quasi-periodic oscillations and (3) chaotic oscillations [3-5,7-10]. When F (the forcing frequency) is close to m/n (a simple rational number) times F N (the naturally oscillating frequency), the neural oscillator usually responds with a n/m synchronized oscillation [7,9,10]. The n/m synchronized oscillation is a periodic oscillation such that n action potentials are generated at locked phases during m cycles of the sinusoidal forcing function. As the fundamental period of the n/m synchronized oscillation just equals m times the period of the forcing function, the stroboscopic plot is composed of m distinct points. Fig. 1 shows stroboscopic plots at multi-phases of a 2 / 3 synchronized oscillation. As each Poincar6 section in fig. 1 is composed of points, the trajectory of the synchronized oscillation is in the form of a limit cycle in the phase space V x dV/dt x S1. Fig. 2 shows stroboscopic plots of a quasi-periodic oscillation. Each stroboscopic plot of the
0375-9601/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
313
Volume 116, number 7
PHYSICS LETTERS A
60 °
30 °
30 June 1986
120 °
90 °
Ikl
150
180"
°
dV/dt
210
14
°
240"
w*t
le 0
270 °
300 °
330"
360"
kO
V Fig. 1. Stroboscopic plots of a 2 / 3 synchronized oscillati#n in squid giant axons (the amplitude A of the sinusoidal force = 4 ~tA, the frequency F of the sinusoidal force = 306 Hz and the naturally oscillating frequency F N = 197 Hz). The number in each stroboscopic plot shows the corresponding phase ( o ) of the sinusoidal force.
30 °
60
©0 °
°
120
°
I
:,"
•
•
'~'~,~ . , . ,
150
180
°
dV/dt
o 1 Ch°
°
'i
.,'?"
.
.
.
, ~v -'~>
240* ::.<. ,.
.d::
!. "1~, .
i
./
/t
=
¢ .i
270 °
330*
300"
360* :.,L
L:;
< . ............i
x-~
~''"
;~
'i
/"
V Fig. 2. Stroboscopic plots of a quasi-periodic oscillation in squid giant axons (A = 0.5 ~A, F = 250 Hz and FN = 203 Hz). 314
Volume 116, number 7
PHYSICS LETTERS A
60"
30"
30 June 1986
90 °
[120" .,.':.
~i~,. ~"
I 50"
dr/or
j#..
180 °
.,,.x...
i
2i0"
,..... '\
240" ¢'.
1.
,,,,;. >. ~..,~
.~,
,~, . f • 4"/
270"
300 ° . r : . , ,-•
330"
360 °
:,
V Fig. 3. Stroboscopic plots of a chaotic oscillation in squid giant axons (A = 2 p,A, F = 266 Hz and F N = 213 Hz).
90"
60"
30"
.r;. y..
120 °
.
, ;-'A
#.,
,~,,,,
1500
1_80"
240"
210" ...,,., -:.:,.
,.j
t270
•
¢.
'
i-
300"
360"
330"
.,',.
:...,•.
.,.
i
.~
V Fig. 4. Stroboscopic plots of a chaotic oscillation in squid giant axons (A ffi 1.5 I~A, F = 270 Hz and F~ ffi 200 Hz).
315
Volume 116, number 7
PHYSICS LETTERS A
merged with another part of the surface (at 150 ° in fig. 4 and at 180 ° in fig. 5) during one period of the forcing function. These processes are similar to those observed in the forced van der Pol oscillator [13,14]. Stroboscopic plots at multi-phases of the forcing function have illustrated that a synchronized oscillation, a quasi-periodic oscillation and a chaotic oscillation in squid giant axons are represented by a limit cycle, a two-dimensional torus and a strange attractor in the phase space V x d V / d t x S 1, respectively. As this method can elucidate the global structures of the attractors, it must be effective for analysing experimental observations of routes to chaotic oscillations such as collapse of tori [7, 10,15-17] in forced systems.
quasi-periodic oscillation asymptotically depicts a closed curve [8,10]. Therefore, the trajectory of the quasi-periodic oscillation is in the form of a twodimensional torus in the space V x d V / d t x S ~. Fig. 3 shows stroboscopic plots of a chaotic oscillation. The stroboscopic plots at the phases 30 ° and 60 ° show that this forced oscillation almost synchronizes to the periodic force at these phases. But the attractor is then stretched (60° 120°), folded (120 ° ~ 2 4 0 °) and compressed (270°--* 360 ° ) during one period of the force. Similar dynamical processes were also reported on the chaotic behavior in the Onchidium giant neuron [6]. The stroboscopic plots at the phases from 150 ° to 240 ° in fig. 3 clearly show that the strange attractor has a structure with multi-layers. Figs. 4 and 5 show another type of chaotic oscillation. While the attractors keep almost tubular structures, a part of the surface seems to be stretched (at 90 ° in fig. 4 and at 120 ° in fig. 5), folded (at 120 ° in fig. 4 and at 150 ° in fig. 5) and
30*
30 June 1986
The authors wish to thank M. Ichikawa for his help in the experiments. A part of this research is financed by the Research Fund of Center for Research at Tokyo Denki University.
90'
60 °
120
°
240
°
36U
°
..'. - .:.
~ H
~7'"
150"
~
...'
,:.,:
i
o¢--~ U-
210
o
°
-.,.. S¢ ~" . a . . ,
dV/dt
•, . ~ ' i . " " -
:270
°
.i
" :'~
. • ~,~*~!~.-"
:300 °
330"
,,,
t
V Fig. 5. Stroboscopic plots of a chaotic oscillation in squid giant axons (A = 2.35 ~A, F = 270 Hz and F N = 1 8 3 Hz).
316
Volume 116, number 7
PHYSICS LETTERS A
References [1] H.R. Hirsch, Nature 208 (1965) 1218. [2] I. Nemoto, S. Miyazaki, M. Saito and T. Utsunomiya, Biophys. J. 15 (1975) 469. [3] A.V. Holden, Biol. Cybern. 21 (1976) 1. [4] R. Guttman, L. Feldman and E. Jakobsson, J. Memb. Biol. 56 (1980) 9. [5] G. Matsumoto, K. Kim, T. Uehara and J. Shimada, J. Phys. Soc. Japan 49 (1980) 906. [6] H. Hayashi, S. Ishizuka, M. Ohta and K. Hirakawa, Phys. Lett. A 88 (1982) 435. [7] K. Aihara, G. Matsumoto and Y. Ikegaya, J. Theor. Biol. 109 (1984) 249. [8] G. Matsumoto, K. Aihara, M. Ichikawa and A. Tasaki, J. Theor. Neurobiol. 3 (1984) 1.
30 June 1986
[9] K. Aihara, G. Matsumoto and M. Ichikawa, Phys. Lett. A 111 (1985) 251. [10] K. Aihara and G. Matsumoto, in: Chaos, ed. A.V. Holden (Manchester Univ. Press, Manchester, 1986). [11] H. Hayashi, S. Ishizuka and K. Hirakawa, preprint. [12] K. Aihara and G. Matsumoto, J. Theor. Biol. 95 (1982) 697. [13] R. Shaw, Z. Naturforsch. 36a (1981) 80. [14] J.M.T. Thompson and H.B. Stewart, Phys. Lett. A 103 (1984) 229. [15] K. Kaneko, in: Chaos and statistical methods, ed. Y. Kuramoto (Springer, Berlin, 1984) p. 83. [16] M. Sano and Y. Sawada, Phys. Lett. A 97 (1983) 73. [17] H.G. Schuster, Deterministic chaos (Physik-Verlag, Weinheim, 1984).
317