proaches the point on the limit cycle P along with the isochron because of K . The relation-
ship between the old phase u at P and the new phase u at P is described below:
u = 1
2p arctan
sin 2pu A + cos 2pu
u + D
RIC
u 2
where D
RIC
is called the phase response curve PRC of the RIC.
In practice, RIC was simulated by the PC, which executed real-time processing of Eq. 1b
Eq. 2. When the RIC’s phase passed across zero, the RIC was considered to have fired. At this
moment, the PC generated a pulse by the DA converter to stimulate the CA3. On the other
hand, when a CA3 bursting was detected, the RIC was stimulated by an impulse with an amplitude
of A. The occurrence of a CA3 bursting was detected when the band-pass-filtered field poten-
tial crossed a threshold level about 0.1 mV. Time series of the CA3 burstings and RIC firings
were recorded under varied coupling modes, each for 10 min, where intensity of the stimulus to the
RIC was changed from A = 0.95, − 0.95 and then A = 0. If one views the RIC as a single pacemaker
neuron, a positive A corresponds to an excitatory coupling, whereas a negative A corresponds to an
inhibitory one Nomura et al., 1994. In the case of A = 0, corresponding to one-way coupling
from RIC to CA3, periodic stimulation with an interval of T
RIC
is realized.
3. Results
3
.
1
. Dynamics of inter-bursting inter6als Fig. 2a – c show the bursting activities of the
CA3 and the firings of the RIC under the varied coupling conditions. The CA3-RIC system exhib-
ited diverse dynamics that were dependent on the coupling modes. First, we investigated the dynam-
ics of the inter-bursting intervals of the CA3 I
CA3
and the RIC I
RIC
Fig. 3, see also Fig. 2d for definitions.
In the case of one-way coupling from RIC to CA3 A = 0, Fig. 3b, I
RIC
remained unchanged while I
CA3
fluctuated significantly interval = 3.49 9 0.67 s, mean 9 S.D. When the stimulus
intensity for the CA3 was increased well above the threshold, CA3 burstings were synchronized to
the RIC firings with a ratio of 1:1 1:1 forced-en- trainment, not shown here. This seemed to indi-
cate that the stimulus intensity used was not sufficiently strong to entrain the CA3 burstings.
Fig. 1. a Schematic representation of the hybrid system consisting of the brain module and the artificial module the CA3-RIC system. Field potentials in the CA3 region were amplified and sent to a personal computer PC, which simultaneously simulated
a nonlinear oscillator RIC. When a CA3 bursting potential was detected by the PC, the RIC was stimulated. On the other hand, the RIC generated an electric stimulus for the CA3 at the moment when the RIC fired u = 0. b Dynamical properties of the RIC
on the phase plane.
Fig. 2. Observed dynamics of the CA3 and the RIC. a – c Field potentials of the CA3 upper and the output sequences of the RIC lower under the varied coupling modes. d Definitions of the inter-bursting intervals of the CA3 I
CA3
and the inter-firing interval of the RIC I
RIC
. e Definition of the time interval c.
When CA3 and RIC were mutually coupled i.e. A 0, fluctuations of I
CA3
seemed to be reduced. As shown in Fig. 3a and d, I
CA3
for A = 0.95 was more stable than the other cases
interval = 2.91 9 0.36 s for A = 0.95 and 3.19 9 0.38 s for A = − 0.95. Data points on the return
map of I
CA3
were shown to mostly concentrate around period = 3 s, but they occasionally devi-
ated from the period. On the other hand, I
RIC
changed frequently between two distinct intervals, about 4 and 5.5 s. Analysis of the return map did
not allow us to determine the switching cycle between the intervals.
In the case of A = − 0.95, RIC fired with pe- riod 2 Fig. 3c and i, which was known from the
return map I
RIC
j + 2 − I
RIC
j consisting of two clusters on the diagonal line not shown
here. In contrast, periodicity of CA3 burstings was not so clear. Nevertheless, by comparing I
CA3
and I
RIC
, we obtained the following integer ratio: mean firing interval of CA3:3.19 s
mean firing interval of RIC:4.83 s :
2 3
. In addition, data points on the return map of
I
CA3
i + 3 versus I
CA3
i were concentrated on a diagonal line not shown. These results suggest
that CA3 and RIC were mutually entrained with a period ratio of 3:2. As we mentioned in the case
of one-way coupling, the stimulus for CA3 was not sufficiently strong for inducing forced-entrain-
ment. However, even with such a weak stimulus, it is possible that CA3 and RIC cooperate to
synchronize their own activities through mutual interactions.
3
.
2
. Dynamics of the latency of CA
3
burstings from RIC firings
In order to determine the relationship between the CA3’s dynamics and the RIC’s ones, we ana-
lyzed the time interval between the RIC firing and the latest CA3 bursting see Fig. 2e for a dia-
gram. Fig. 4a basic plot of c shows that c’s dynamics are seemingly complex in the case of
A = 0.95. On the corresponding return map Fig. 4d, one can clearly recognize five clusters. Since
the clusters approximately form a single-valued function, the complex dynamics would be re-
garded as deterministic. In the case of A = − 0.95, c’s dynamics are almost period 3 because
the return map consists mainly of three clusters Fig. 4f. This result also supports 3:2 mutual
entrainment between the CA3 and the RIC. In contrast, c’s dynamics for A = 0 seem to be
quasi-periodic Fig. 4b. Nevertheless, as shown in Fig. 4e, distribution of the data points on the
return map are qualitatively similar to the case of A = − 0.95.
3
.
3
. Modeling the phase response cur6e of CA
3
To model the phase response curve of the CA3, we explored the stimulus-to-response relationship
of the CA3 to understand the dynamics of c. We assumed the CA3 as a nonlinear oscillator, be-
cause it generated bursts repetitively with an ap- proximately fixed period T
CA3
if no stimulus was applied. In addition, even if the bursting interval
was changed by some stimulus, the interval was restored by itself. Therefore, we investigated the
phase response curve PRC of the CA3, which is an analyzing tool of internal structure of a nonlin-
ear oscillator.
Fig. 5 shows the relationship between the burst- ing interval T and the stimulus timing d, which
were obtained from the data presented in Fig. 3. As shown in the figure, overall response charac-
teristics were consistent through the coupling modes. Regardless of the coupling modes, the
bursting interval Td changed dramatically, de- pending on d. That is, if a stimulus was applied
within T
R
after the last CA3 bursting d B T
R
, the stimulus did not affect the bursting interval.
In contrast, if a stimulus was applied after the refractory period T
R
, CA3 generated a burst im- mediately T
D
after the stimulus. We estimated T
CA3
, T
R
, T
D
as 3.10, 1.50, 0.10 for A = 0.95,
Fig. 3. Dynamics of the CA3-RIC system under the varied coupling modes. a – c Basic plots of the inter-bursting intervals of CA3 I
CA3
, and the inter-firing interval of the RIC I
RIC
, versus event time. d – f Return maps of I
CA3
corresponding to the data a – c. g – i Return maps of I
RIC
corresponding to the data a – c.
Fig. 4. Return maps of the time-interval c under the varied coupling modes. a – c Basic plots of c. d – f Return maps of c. Dots indicate the experimental data. Thick lines are the reconstructed return maps; thin lines represent the stable orbits allowed in the
reconstructed return maps. The entrainment ratio for the stable orbit is indicated beside the orbit.
3.70, 1.80, 0.10 for A = 0, and 3.42, 1.80, 0.10 for A = − 0.95 using the least-square method.
Based on these results, we modeled the CA3’s PRC D
CA3
as follows Fig. 5e:
where f = dT
CA3
is the normalized ‘phase’ of the CA3 ranging between 0 and 1.
3
.
4
. Reconstructed return map of c based on the phase response cur6es
If the CA3’s PRC fully describes the dynamical properties of CA3 under the varied coupling con-
ditions with RIC, the dynamics of c could be reconstructed based on the PRCs of both oscilla-
tors. Conversely, if there were disparities between the experimentally obtained and the reconstructed
return maps; they indicate a nature of the CA3- RIC system’s dynamics that could not be de-
scribed within the framework of a coupled two-oscillator system.
The reconstructed return maps of c are pre- sented in Fig. 4d – f together with the experimental
data. The mathematical expression of the maps is described in Appendix A. As shown in the graphs,
the reconstructed maps almost agree with the experimental data. The stable orbits for the recon-
structed maps indicate that the CA3 and RIC would mutually entrain with period ratios of 11:7
for A = + 0.95, and of 3:2 for A = 0 and for A = − 0.95. Regardless of coupling conditions,
the reconstructed orbit almost covers the experi- mental data. Therefore, the actual orbit is consid-
ered to wander in the neighborhood of the stable orbit predicted by the reconstructed map. Never-
theless, there are deviations between the recon- structed
map and
the experimental
data, especially in the case of one-way coupling, which
is depicted by an oval in Fig. 4e. We believe the deviation indicates fluctuations in the intrinsic
bursting period T
CA3
just after the CA3 bursting immediately evoked by the stimulus from the
RIC, because we know the concrete cluster is
D
CA3
f = T
CA3
− TT
CA3
f T
CA3
= 0,
0 5 f B T
R
T
CA3
T
CA3
− T
D
T
CA3
− f
otherwise,
Fig. 5. Stimulus-to-response characteristics of the CA3 under the varied coupling modes. a – c Plots of inter-bursting interval T as a function of the stimulus timing d. d Definition of the inter-bursting interval T and the stimulus timing d. e Modeled phase
response curve of the CA3 D
CA3
.
constructed by the point cn, cn + 1 = T
D
,T
CA3
+ T
D
according to the theory see Appendix A. Actually, the fluctuation was
significantly large in the case of one-way coupling. These
results suggest
that the
dynamical properties
of fluctuation
in T
CA3
changed dependent on the coupling mode.
4. Discussion
