 |
INTRODUCTION |
Electrical coupling is a general property of the developing
(Pienado et al. 1993
; Walton and Navarrete
1991
) and adult CNS (Galarreta and Hestrin
1999
; Gibson et al. 1999
; Kandler
and Katz 1995
), where it tends to enhance and
synchronize the activity of coupled neurons. The efficacy of electrical
coupling may be altered by both ontogenetic and modulatory processes.
For example, several studies have demonstrated that gap junctions are
abundant early in development and then decline during subsequent
maturation (Kandler and Katz 1995
; Pienado et al.
1993
). In the adult CNS, moreover, the strength of electrical
coupling between neurons may be substantially altered by various
transmitters or hormones (Baldridge et al. 1998
;
Hatton and Yang 1996
; Johnson et al.
1993
). Such alterations, either in the expression of gap
junctions or via their modulation, have potentially important
functional implications for neuronal network operation. However,
although electrical coupling has been studied extensively at the level
of cell-to-cell communication within a given network (Fukuda and
Kosaka 2000
; Gibson et al. 1999
; Graubard
and Hartline 1987
; Tamas et al. 2000
),
little is known about its role in the interaction between different
functional networks during ontogeny.
To address this problem we have explored the potential role of
electrical coupling in the maturation of neural networks in the lobster
stomatogastric nervous system (STNS). In the adult animal, this system
controls rhythmic food-processing movements of the foregut organized in
three different regional behaviors. The underlying motor patterns are
generated by three discrete networks that consist of endogenous
oscillators interconnected by graded inhibitory synapses
(Harris-Warrick et al. 1992
; Selverston and
Moulins 1987
) and that operate at completely different
cycle frequencies (Casasnovas and Meyrand 1995
;
Meyrand et al. 1991
). By contrast, in the lobster embryo
the same population of STNS neurons generates a coordinated activity
organized in a single motor pattern (Casasnovas and Meyrand
1995
; Fénelon et al. 1998
; Le Feuvre et al. 1999
). However, recent in vitro
experiments have shown that adult-like network properties are already
embedded in this embryonic network but are continuously masked by an
unknown mechanism (Le Feuvre et al. 1999
). Since
electrical coupling is prevalent in the embryonic nervous system then
decreases during ontogeny (Kandler and Katz 1995
;
Pienado et al. 1993
), we have explored the possibility
that diminishing gap junction coupling may be responsible for the
developmental emergence of adult STNS networks.
To address this issue we have used a mathematical model to perform the
reverse experiment in which electrical coupling is introduced within
and between three adult-like model STNS networks. The model networks
generated antiphasic oscillations due to endogenous oscillatory
properties of individual cells and reciprocal inhibition within each
network. Each model cell consisted of a generalized van der Pol
relaxation oscillator. Since spikes are not necessary for the
generation of membrane oscillations and STNS motor patterns (Raper 1979
), model membrane potential
oscillation was produced without spikes. A network constructed by
connecting such model cells via instantaneous graded synaptic
transmission captures essential features of the biological STNS network
(Rowat and Selverston 1993
). Gap junctions were modeled
as giving currents proportional to the differences in voltage of any
two coupled cells.
We find that the addition of widespread electrical coupling into the
STNS model can unify the activity of all cells belonging to the
different networks into a single multiphasic pattern. Concomitant with
this network reorganization is a dramatic decrease in the oscillation
amplitude of all neurons. Importantly, both phenomena are reminiscent
of the activity of the single STNS embryonic network. Thus our data
suggest that rather than a progressive maturation of cellular and
synaptic properties within a given network, a gradual reduction of
electrical coupling strength may account for the emergence of
preexisting adult network properties during ontogenesis.
 |
METHODS |
Dissections and electrophysiological study of the STNS of the
lobster Homarus gammarus were performed as previously
reported in the adult (Meyrand et al. 1994
) and in the
embryo (Casasnovas and Meyrand 1995
).
In our simulation analyses, a single-cell model was used that was a
modified relaxation oscillator generating membrane potential oscillations by slow (Islow) and fast
(Ifast) voltage-dependent membrane
current (Rowat and Selverston 1993
). When incorporated into a network, the model neuron also contained synaptic
(Isyn) and electrical junction
(Iej) currents. This dimensionless
model was written in two differential equations:
1) A total current conservation
where
was cell membrane potential and
m was membrane time constant;
2) A slow current activation
where
S was the time constant and
S the steady-state conductance of the slow
current. Fast current, combined from a leak and fast voltage-dependent
currents, was n-shaped to a degree determined by 1
f corresponding to the slope of the
instantaneous current-voltage (I-V) curve at the origin, as
given by Ifast =
tanh
(
f
).
Synaptic current was modeled as
where w was maximal synaptic conductance,
f(
pre) = (1 + e
4
pre)
1
determined a fraction of postsynaptic channels open due to graded transmitter release,
pre was the potential of
the presynaptic cell, and Epost was
the synaptic reversal potential. Gaussian noise
(t),
restricted to a very low amplitude, was introduced to synaptic
transmission to make cells of identical membrane and synaptic
properties slightly different; otherwise a transition from synchronous
to antiphase oscillations during a decrease in g did not
occur. Iej was modeled as described in
the text. In the experiment shown in Fig. 8, networks were constructed
from Burster C neurons (Epstein and Marder 1990
).
Network cells differed in their values of Ca2+
and Ca2+-activated K+
conductances and in the rate of accumulation of free
Ca2+, all of which were larger in the fast than
in the slow network. Synaptic current was modeled as
Isyn = gsynS(
pre)(
syn) (Skinner et al. 1994
),
where S(
pre) = 0.5(1 + tanh
[(
pre
thresh)/
slope)]),
syn =
80 mV,
thresh =
40 mV, and
slope = 15 mV. Simulations were
run using the software XPPAUT by B. Ermentrout.
 |
RESULTS |
Motor patterns generated by the adult and embryonic lobster STNS
In adult H. gammarus, the isolated STNS generates three
spontaneous motor patterns (gastric, esophageal, and pyloric rhythms) with different cycle frequencies (Fig.
1A). The fastest rhythm (cycle
period 1-2 s) is expressed by the pyloric network while the esophageal
and gastric networks generate rhythms of intermediate (4-6 s) and slow
(10-15 s) periods, respectively. By contrast, in the embryo the same
neuronal population (Fénelon et al. 1998
) generates coordinated activity organized in a single motor pattern that
drives target muscles at the same cycle frequency but with different
phases (Fig. 1B) (Casasnovas and Meyrand
1995
; Le Feuvre et al. 1999
). In addition to
these different temporal relations in the adult and embryonic STNS
rhythms, the amplitude of oscillations in individual embryonic neurons
is significantly lower than in their adult counterparts. This is
evident in Fig. 1C, where the same pyloric cell type, the
lateral pyloric (LP) neuron was recorded intrasomatically in the adult
and embryonic STNS. In both cases the cells express membrane potential
oscillations underlying bursts of spikes. However, the amplitude of
oscillations in the adult neuron is considerably greater than in the LP
cell of the embryo. Moreover, a significant difference in oscillation
amplitude between embryonic and adult cells appears to be a general
feature of the STNS neuronal population (see also Casasnovas and
Meyrand 1995
) as seen in Fig. 1D, which shows
amplitude distributions of a variety of embryonic and adult neurons
recorded from the pyloric and gastric networks in the stomatogastric
ganglion [mean ± SE amplitude of oscillation; adult, 34.9 ± 10.6 mV (n = 58 neurons); embryo, 6.5 ± 3.7 mV
(n = 41 neurons)].

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Fig. 1.
Rhythmic activity of stomatogastric nervous system (STNS) motor
networks in adult and embryonic lobster, Homarus gammarus.
A: different spontaneous rhythms recorded extracellularly and
simultaneously in vitro from motoneurons innervating pyloric (Pyl),
esophageal (Oeso), and gastric (Gast) muscles in adult. Pyloric
activity recorded from the pyloric dilator nerve, esophageal activity
from the ventral posterior esophageal nerve, gastric activity from the
anterior gastric nerve. B: single multiphase rhythm recorded
intracellularly in stage E98% embryo from pyloric, esophageal, and
gastric muscles. C: large-amplitude membrane potential
oscillations underlie bursting in an adult pyloric lateral pyloric (LP)
neuron, while in the embryo, the same neuron type expresses small
membrane potential oscillations. Maximal hyperpolarized levels are
indicated. D: pooled amplitudes of spontaneous membrane
oscillations in gastric and pyloric neurons in adult and embryo. In
both the adult and embryo, the amplitude of oscillations were measured
from 58 and 41 cells, respectively, from 25 embryonic and 25 adult
stomatogastric. The amplitudes were measured from the lowest membrane
potential level to the peak of the slow wave. Horizontal bars: 1 s. Vertical bars: 10 mV.
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Effects of altering internetwork electrical coupling in a STNS
model
To test the hypothesis that electrical coupling may be involved in
masking adult-like network properties in the embryonic STNS, we
interconnected three separate model networks, each consisting of two
mutually inhibitory oscillatory neurons, through gap junction coupling
(see METHODS). Each neuron was coupled through the same junction conductance g to all other neurons except its
network partner (Fig. 2A).
Membrane and synaptic parameters were initially set to the same values
in all cells except the slow current, which determined the different
intrinsic frequencies of the networks (Fig. 2B, left).

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Fig. 2.
Effects of electrical coupling between 3 model oscillatory networks.
A: schematic representation of the model system studied.
Each network is constructed with identical neurons linked by reciprocal
inhibition of the same strength (black dots). Bars between the cells on
the right indicate pattern of gap junction coupling.
B: a large step coupling ( g = 0.45 at
arrow) transforms independent rhythms into damped synchronized
activity. C: cell traces as in B on slower time
scale and superimposed on the same baseline level. Stepwise coupling
increase (g, 0 to 0.5; g, steps = 0.05)
gradually dampens oscillations until rhythm unification occurs at
g = 0.45 (left arrow). Reducing coupling
(g, 0.5 to 0) unmasks different network frequencies at
g = 0.3 (right arrow) and thereafter oscillation
amplitudes increase. In B and C,
m = 0.1666, S = 5, Af = 1, Epost = 4,
f = 2 for all cells,
S = 1.8, 4, 10 for slow, moderate, and fast
oscillators, respectively. Synaptic strength w = 0.2.
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|
When electrical coupling was set to zero (g = 0, Fig.
2B), the model networks expressed basic features of adult
STNS output consisting of three independent rhythms with different
frequencies (cf. Fig. 1A) and strict alternation between
members of each network. Introduction of strong coupling between these
networks (from g = 0 to g = 0.45)
produced three dramatic changes in their activity (Fig. 2B,
right). First, the oscillation amplitude in all neurons was
abruptly decreased by ~90%. Second, neuronal oscillations within
each network switched from alternation to synchrony. Third, the
different network rhythms became unified into a single synchronous rhythm. The evolution of these rhythm alterations and the damping of
oscillations is illustrated in Fig. 2C, where all voltage
traces as in Fig. 2B are superimposed on the same baseline
at a slower time scale. Here a stepwise increase, then decrease, in
coupling strength (
g = 0.1) produced a gradual and
reversible change in oscillation amplitude of all cells. A switch
between different (shaded areas) and unified (unshaded area) rhythms
occurred when coupling reached critical values (Fig. 2C, see arrows).
To explore the robustness of the above phenomena, we tested the
response of the model to an alteration in the strength of intra-network
inhibition that remained equally distributed among the networks (Fig.
3). In contrast to the experiment
presented in Fig. 2C, where we observed the dynamic
alterations of the activity pattern in response to changing
g, here we tested the behavior of the system after it
reached steady state. When no inhibition was present in the system
(w = 0) and neurons were set to oscillate in anti-phase
within each network, increasing g led quickly to rhythm
synchronization (transition from shaded to unshaded areas) that was
accompanied by a slight decrease in oscillation amplitude (solid
lines). The more inhibition was introduced into the system, the more it
resisted the synchronizing influence of electrical coupling. Indeed, as
inhibitory strength w was increased (Fig. 3A,
w = 0.15, 0.2), network synchronization occurred at
progressively higher coupling strengths and with a correspondingly
larger decrease in oscillation amplitude before final rhythm synchrony
occurred. If inhibition was too strong (Fig. 3A,
w = 0.3), increasing g now failed to produce
network unification with synchrony and, instead, led to a complete
collapse of activity. In this state, however, a single rhythm appeared
when neuronal oscillatory capability was strengthened by increasing the
fast conductance value (not shown).

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Fig. 3.
Evolution of oscillation amplitude and frequency as a function of
electrical coupling strength for different levels of intranetwork
inhibition. A: mean oscillation amplitude vs.
g in similar experiment as in Fig. 2C,
for different indicated synaptic strengths (w). Shaded
areas indicate multiple rhythms, whereas unshaded areas indicate single
rhythmic activity. Note at w = 0.3, no
synchronization occurred. Data from moderate frequency network
calculated from 100 time units simulation. B: with
increasing g, 3 different network periods (shaded area)
gradually converge onto a common value (unshaded area), which is
similar for different inhibitory strengths (horizontal bar;
wmin= 0.0, wmax = 0.2). This phenomenon is
reversible. Cell parameters as in Fig. 2.
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|
The damping of oscillations was accompanied by a change in cycle period
of all three networks until they converged onto a common value as
illustrated for w = 0.15 (Fig. 3B). The
period of the synchronized rhythm was very weakly dependent on the
strength of inhibition and decreased slightly with increasing
w, as indicated by the upper (w = 0.0) and
lower (w = 0.2) limits within the horizontal shaded
area in Fig. 3B.
It is noteworthy finally that some hysteresis accompanied these changes
in electrical coupling. This was due not only to experimental procedure
(i.e., a stepwise change in coupling strength before the system had
reached steady state; Fig. 2C), but also when a steady-state
behavior was being tested (Fig. 3B). This in turn suggests
that over a certain range of electrical coupling, a bi-stability may
occur where both different or synchronized rhythms can provide a stable solution.
Network potential (NP) as a representation of network activity
We next sought an explanation for these damping and unifying
effects of electrical coupling between different oscillatory networks.
Initially, consider two distinct entities, a single neuron a
and a single network N that consists of n cells.
Let neuron a be uniformly coupled through conductance
g to each of n cells of network N. The
junction current between a and N is then given by
|
(1)
|
where
a is the membrane potential of
a and
1,
2, ... ,
n are
the network cell membrane potentials. Equation 1 can be
written as
|
(2)
|
which gives
|
(3)
|
Equation 3 thus describes the junction current of
neuron a coupled through conductance ng to the
mean membrane potential of all cells in network N. We will
call this variable "Network Potential" (NP) as it is a dynamic
image of the activity of all the cells of network N that
interact with cell a. Thereby
|
(4)
|
As seen in Fig. 4, the global
operation of a given neural network is encoded in its NP. For instance,
consider two model neurons that oscillate in phase opposition due to
reciprocal synaptic inhibition. When w is set to 0, the
voltage traces are perfectly symmetrical, and the NP generated is thus
a tonic signal situated at mid-amplitude of oscillation (see arrows
Fig. 4A). This is due to the fact that in our model
oscillatory cell, voltage traces situated above and below level
= 0 (which is the resting potential of the cell and a mid-line
of membrane potential oscillation) are mirror images. Therefore for two
cells that oscillate in anti-phase, the mean value of their membrane
potentials is
= 0. However, for w > 0, the NP
now becomes a small-amplitude phasic signal due to a slight asymmetry
caused by synaptic inhibition within each half cycle (Fig.
4B). NP is also shaped by other network properties, as for
example, when the oscillatory capability of one neuron is weakened
(Fig. 4C) or a third cell is added to the network (Fig.
4D).

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Fig. 4.
Generation of different Network Potentials (NPs). A and
B: 2 identical oscillatory neurons interconnected by
inhibitory synapses of identical strength (w).
A: for w = 0 a tonic NP is
generated. B: for w = 0.2 the
network generates a small amplitude phasic NP with a frequency twice as
high as the inherent frequency of the network. C and
D: asymmetry in cell membrane properties due to an
increase of fast current conductance in one cell (C) or
an increase in cell number (D) leads to large amplitude
NPs with different shapes. Arrows indicate equivalent potential levels
(V = 0) for all traces. In A-D,
f = 2 except f = 0 in neuron a
in C; S = 10; in A,
w = 0; B and C,
w = 0.2; in D,
wa, wb = 0.1, wc = 0.2. Other parameters as in
Fig. 2.
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|
With this understanding, we then examined how the NPs generated by two
networks A and B are involved in their electrical interaction. Again
networks A and B consisted of two model cells a/b
and c/d, respectively, which were linked by
reciprocal inhibition within each network while any two cells belonging
to different networks were coupled via electrical conductance
g (Fig. 5A). The
coupling current ia between cell
a and cells c and d
of network B (Fig. 5B) is given by
|
(5)
|
Thereby
|
(6)
|
where, according to Eqs. 3 and 4,
NPB = (
c +
d)/2. Equation 6 describes the
interaction between cell a, represented by its potential
a, and network B, represented by its
potential NPB (Fig. 5C). If we now add
the contribution of cell b to the internetwork communication
(Fig. 5D), which is described by the current
then the total current I between A and B is given by
|
(7)
|
which leads to
|
(8)
|
Equation 8 thus describes the electrical relationship
between networks A and B through the interaction of their network
potentials NPA and NPB,
respectively (Fig. 5E). Similarly, the total junction current I, through which any two networks A and B, coupled
in an "all-to-all" fashion, communicate, is given by
|
(9)
|
where G = nAnBg is
the internetwork coupling strength,
nA,
nB, and NPA,
NPB are numbers of cells and NPs of A and B,
respectively.

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Fig. 5.
Modeling NP. A: cells belonging to reciprocally
inhibitory networks A and B are coupled
in an "all-to-all" fashion via identical coupling strength
g. B: a sub-circuit of the model
presented in A consisting of single cell
a and network B. a, c, and
d are membrane potentials of cell a,
c, and d, respectively. Note anti-phase
oscillations in network B. C: cell a
"senses" network B through the latter's Network Potential
NPB, which represents the mean membrane potential of
network members c and d. The interaction
between a and NPB is mediated by electrical
coupling of the strength 2g. D: cell
b interacts with network B in a similar fashion as cell
a. E: the activity of network A is represented by its
Network Potential NPA (similar to the construction of
NPB in C), which now interacts with
NPB via coupling strength 4g.
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|
Equation 9 is also valid for nonuniform coupling. In this
case
|
(10)
|
where a1,
a2, ... , an
specify the proportion of internetwork coupling G provided
by individual network cells (ai = 1/n for i = 1, 2, ... , n
in the case of uniform coupling).
It is important to note that the NP as defined by Eq. 10 is
a representation of a given network in terms of a particular
distribution of its electrical coupling with another network. For an
all-to-all uniform coupling, this variable is equivalent to the mean
value of the membrane potential of all cells belonging to a given
network, as in Eqs. 4-9. However, in a general case the
contribution to NP of individual network members may be different
(i.e., ai, i = 1; n
in Eq. 10 may be different). Thus a given network may be
represented by different NPs in its interaction with different
networks, depending on the distribution of electrical coupling with
these networks.
Electrical interaction between one cell and a network
To investigate the essential mechanism underlying the damping and
unifying effects of electrical coupling, we first analyzed the
sub-circuit of the model illustrated in Fig. 5, containing single cell
a with membrane potential
a and
network B, built of two reciprocally inhibitory oscillators that
generates potential NPB (Fig.
6A). As seen by Eq. 6, with increasing coupling g, the coupling current
will equalize the potentials
a and
NPB. Assume now that network B is solely a
receiver of the potential
a of cell
a, without any reciprocal influence (Fig. 6A).
Therefore as
a is not affected by this
interaction, NPB will be gradually sculpted into
a's own waveform (compare a and
NPB traces in Fig. 6A). These
modifications in NPB reflect the alterations in
the intranetwork coordination that is changed from anti-phasic
oscillation to full synchrony (see traces c and d
in Fig. 6A). It must be noted that the weaker the reciprocal
inhibition within the network, the lower is the coupling strength at
which synchrony can be imposed on network B members by cell
a.

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Fig. 6.
Interaction between a single cell and a network. A:
assuming cell a influences network B without reciprocal
interaction. Increasing g (from g = 0 to 1.2) results in a gradual unification of NPB with
a (cf. 2 top traces). As a result,
NPB amplitude increases significantly, reflecting the
increased level of synchronization between cells c and
d of network B. B: assuming network B
influences cell a unidirectionally. Due to a low
amplitude of NPB, cell a undergoes a
significant damping of oscillation amplitude (cf. top
traces for g = 0, 0.3, 0.6) as a result of
unification of a with NPB. C:
when reciprocal interaction is permitted, both damping and
synchronization of all cells occur as a final state with increasing
g (right panel). At intermediate coupling
strength, rhythm unification with patterned activity, instead of full
synchrony, occurs (middle panel). In
A-C, f = 2, s = 1.8 in a, s = 5 in c and d,
wc, wd = 0.2. Other parameters as in Fig. 2.
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|
In a reverse case of such unidirectional influence, where network B now
affects cell a,
a will be
gradually modified into the form of NPB (Fig.
6B). Since the network potential of two oscillatory cells is
a low-amplitude signal situated close to the mid-line of their
oscillations (i.e., toward
= 0, see Fig. 4B), the
major effect of this NPB influence on cell
a will be a significant damping of its oscillation (Fig.
6B). A decrease in the oscillation amplitude of cell
a as a function of the strength of electrical coupling can
be estimated by assuming NP = 0. Such an approximation is also
applicable in the case when a cell is electrically coupled to a large
population of cells oscillating at very different phases (see
APPENDIX).
In a final step, we allowed for a reciprocal influence between
cell a and network B. In this case the two previously
described effects of electrical coupling, i.e., damping and
synchronization, are present together (Fig. 6C). Thus with
increasing g the system reaches the state where all cells
oscillate in synchrony and with diminished amplitudes. However, the
damping effect is not as strong as in the case of a unidirectional
influence between network B and cell a (Fig. 6B),
since NPB now increases in amplitude due to the
synchronization of network B members imposed by cell a. Significantly, unified patterns with multiple phases may be expressed at moderate coupling strengths (Fig. 6C, middle
panel). In this case, the rhythm of cells a,
c, and d are already unified, whereas cells
c and d, due to their reciprocal inhibition are
not yet fully synchronized. It is important to reemphasize here that we use the term unification with pattern, and not synchronization, to
describe this new organization of neuronal activity into a common cycle
frequency. In such unified rhythms, individual cells can still exhibit
multiphasic coordination without full synchronization, as in the case
of the motor pattern expressed by the embryonic STNS network (Fig.
1B) (see also Casasnovas and Meyrand 1995
).
Until now we have examined the effect of electrical coupling between a
single cell and a given network in which reciprocal inhibition
generates antiphasic activity. In these conditions, whereas the
oscillatory cell expresses a large-amplitude signal, the network
generates a low-amplitude NP. As a result of increasing g,
therefore the potential of cell a will be damped while the NP will be enhanced (see unification of
a and
NPB through the coupling current, Eq. 6). This corresponds to a change from anti-phasic to synchronous
coordination of network members. The capacity to enhance the amplitude
of NP through electrical coupling and thereby increase the level of
synchrony within the network is dependent on the strength of
intranetwork inhibitory connectivity, which acts in opposition to
electrical coupling.
Electrical interaction between two networks: effect of
heterogeneity
The phenomena described above concerning the communication between
a single cell and a network are also expressed in the interactions between two networks in which the neurons are linked by inhibitory synapses. As demonstrated in Fig.
7A, uncoupled networks A and B
generate different NPs (Fig. 7A, left, top 2 traces) due to their different inherent cycle frequencies. The
amplitude of the NPs is low because of the lack of heterogeneity in the
networks that are built of identical cells, interconnected by synapses of identical strengths (cf. Fig. 4B). As coupling is
introduced, each cell becomes influenced by the NP of the neighboring
network as described above. This produced damping of oscillation
amplitude in each cell (compare cells a-d from
left to right panels in Fig. 7A).
Additionally, changes in coordination within each network occurred: at
intermediate g the network rhythms expressed 1:2 coordination (middle panel, Fig. 7A), whereas for
large g the rhythms became unified (right panel,
Fig. 7A). These changes in coordination were accompanied by
a gradual unification of the two NPs as described by Eq. 9.

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Fig. 7.
Networks communicate through their NPs. A-C: 2 uncoupled networks generate independent NPs (left, top 2 traces). With increasing g (0, 0.4, 0.8),
interacting NPs become unified and entrain the entire neuronal
population into a single, fully synchronized rhythm (right
panel, neuron traces). A: if the cells in both
networks are interconnected by identical synapses and NPs are of low
amplitude, rhythm unification occurs together with synchronization of
neurons only for g = 0.8. B:
weakening synaptic strength in network A results in rhythm unification,
cells a and b being synchronized for
g = 0.4, while cells c and
d continue to oscillate in anti-phase. C:
with intranetwork heterogeneity in synaptic strength, networks generate
large-amplitude NPs. As a consequence, multiphasic unification of both
networks occurs at g = 0.4. Note eventual damping
of activity (at g = 0.8) is weak
(B), moderate (A), or complete
(C), depending on the strength of inhibition in the
system. In A-C, f = 2, S = 1.8 in a and b,
S = 5 in c and d. In
A, wa-d = 0.2; in
B, wa,
wb = 0.1, wc, wd = 0.2; in C, wb,
wd = 0.05, wa, wc = 0.3. Other parameters as in Fig. 2.
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|
To test the effect of heterogeneity on the behavior of this system, we
first weakened the synaptic strength within network A (Fig.
7B). Such an alteration reduced the resistance of network A
to the synchronizing influence of incoming NPB.
As a consequence, cells a and b belonging to
network A now became fully synchronized at intermediate coupling
strength (g = 0.4, Fig. 7B). Thereby, the
amplitude of NPA was significantly enhanced,
providing a strong signal that entrained network B and reduced the
damping effect on this network. However, since reciprocal inhibition in
network B is stronger than in network A, the cells c and
d were still not yet fully synchronized (see middle
panel, Fig. 7B). Thus in contrast with the previous
case (Fig. 7A), the cells now expressed a unified rhythm
with multiple phases at the moderate coupling strength
(g = 0.4). At higher coupling strength
(g = 0.8), the cells within each network expressed
fully synchronized coordination as previously, but with a relatively
high oscillation amplitude. This was due to the reduced amount of
inhibition within the system (compare A and B in
Fig. 7).
Second, we introduced intranetwork heterogeneity in the strength of
synaptic inhibition in both networks (Fig. 7C). These changes enlarged the amplitude of corresponding NPs (compare 2 top traces in Fig. 7, A-C, left panels) and thereby
reinforced the entraining effect of a given NP on the members of the
other network. Thus the network rhythms already became unified at
moderate coupling strengths, with individual cells still expressing
out-of-phase coordination maintained by inhibitory coupling
(middle panel, Fig. 7C). A further increase in
g now led to a complete collapse of oscillations. This was
again due to the presence of strong inhibition in the system (compare
Fig. 7, C and B; see also
DISCUSSION).
Thus we have shown that the unified rhythm can express different
temporal patterns. With increasing g, the system always
attains full synchrony (see right panels in Fig. 7,
A and B) or collapse (see right panel
in Fig. 7C), depending on the strength of inhibitory connections within the networks (see also Fig. 3A). However,
at intermediate g patterned rhythm unification may occur
before either of these final states is reached (see middle
panels, Fig. 7, C and D). The necessary
condition for such multi-phasic unification is that the incoming NP is
sufficiently large to entrain the neighboring network members to its
own frequency, whereas the reciprocal inhibition prevents them from
reaching full synchrony.
The generality of the phenomenon described above was further
tested by experiments with more complex neuron models containing five
conductances (Epstein and Marder 1990
). Here again as
for the simpler neuron model used above, increasing coupling led to damping of oscillations and network unification with multiple phases
(Fig. 8). Interestingly, in this model
where neurons generate fast spikes, introducing electrical coupling can
result in damping of oscillation amplitude below spike generation
threshold. Nonetheless, the unified multiphasic patterns of slow
membrane potential oscillations still occur.

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Fig. 8.
Interaction between networks built from spiking model oscillators with
5 currents. Increasing g (0, 0.03, 0.037) results in NP
unification and a damped single rhythm. Note 1:2 network coordination
despite 1:1 NP coordination for g = 0.03. Synaptic
conductance (in mmho/cm2)
gsyn(a),
gsyn(b) = 0.1, gsyn(c),
gsyn(d) = 0.02; membrane conductance
(in mmho/cm2)
gCa = 0.12 in a and
b, 0.18 in c and d,
gKCa = 0.32 in a and
b, 0.6 in c and d;
electrical conductance g as indicated in
B (in mmho/cm2); Ca2+
accumulation rate (in ms 1) = 0.003 in
a and b, 0.005 in c and
d. Other parameters as in Epstein and Marder
(1990) .
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Effect of widespread electrical coupling in a more realistic STNS
model
In a final step we tested a more realistic model of the
adult STNS, which consisted of two two-neuron circuits (corresponding to esophageal and gastric network subsets) and one three-neuron network
(equivalent to the pyloric network) operating at different frequencies
(Harris-Warrick et al. 1992
). In an approximation to the
general early developmental situation where electrical coupling is
widespread and nonspecific (Kandler and Katz 1995
; Pienado et al. 1993
; Walton and Navarrete
1991
), we introduced electrical coupling within these model
networks, in addition to internetwork coupling (Fig.
9A, top). Both
damping and unification with multiple phases persisted under such
experimental conditions (Fig. 9A, bottom). Indeed,
electrical coupling between cells belonging to the same network
produced additional damping of membrane potential oscillations and
effectively weakened their reciprocal inhibition. This in turn
facilitated the unifying influence of incoming NPs (not shown). As seen
in Fig. 9B (see also simulations in Figs. 2B and
3) escape from coupling revealed three independent network rhythms
corresponding to the intrinsic frequency of each circuit. Moreover, as
illustrated in the simulation of Fig. 9C, these networks can
be reunified, but without damping (Fig. 9C, bottom), if a number of specific internetwork electrical connections is preserved (Fig. 9C, top). In this case the single-cell activity
constitutes the NP of a given network and thereby represents the
network in its communication with the other networks. Importantly,
since this NP signal is now of large amplitude, there is no damping effect on individual cell oscillations as in the case of widespread electrical coupling (cf. Fig. 9, A and C).

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Fig. 9.
Effects of modulating electrical coupling on the activity of a network
ensemble. Shown are 2- and 3-cell networks and the activity patterns
they produce when nonspecifically coupled (A), uncoupled
(B), and specifically coupled (C). Note
emergence of large-amplitude independent network oscillations from a
damped single rhythm (A and B), then
unification without damping (B and C).
The boxes highlight the periods of the different network outputs.
f = 2 except f = 1.5 in
b and d, 2.5 in e,
S = 1.8 in a and b, 4 in c and d, 10 in
e-g; wa,
wc, wgf = 0.05, wb, wd,
wfe, wfg,
wge = 0.2, we = 0.3. Other parameters as in Fig.
2.
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DISCUSSION |
The results described here indicate that widespread electrical
coupling dampens neuronal oscillations and can fuse different network
outputs into a single multiphasic pattern. The fact that strong
electrical coupling unifies the cycle frequencies of different oscillators comes as no surprise to any neurophysiologist. Also, despite the classical view of gap junctions as a synchronizing mechanism, it is already known that in some circumstances such coupling
can actively foster asynchrony in a population of spiking cells
(Chow and Kopell 2000
; de Vries et al.
1998
; Sherman and Rinzel 1992
). Indeed,
depending on spike shape/size and gap junction strength, coupled
neurons can be synchronized, anti-synchronized, phase locked at any
phase, or lose their ability to fire periodically (Chow and
Kopell 2000
). This can be understood by thinking of electrical
coupling as acting as excitation or inhibition, depending on whether
the electrically coupled cell partner is active in the spike phase or
in the postpolarization phase, respectively. We show here that in a
population of mutually inhibitory cells expressing slow membrane
potential oscillations, there exists another mechanism for multiphasic
unification of neuronal activity through electrical coupling.
Specifically, this phenomenon is a consequence of a battle between two
opposing forces: one of them a synchronizing influence of electrical
coupling, and the other, mutual inhibition that tends to maintain cell
activity in antiphase and thereby competes directly with the action of electrical coupling. When electrical coupling is weak, cells within each network oscillate in anti-phase due to inhibitory coupling and
network rhythms are independent. When electrical coupling is strong,
all cells belonging to multiple networks are forced to oscillate in
phase. However, for intermediate coupling strengths, a third state may
occur. In this state the network rhythms are already unified due to
interaction of network potentials conveyed through electrical
junctions, whereas phase differences between cells still occurs due to
the presence of inhibition, which is still relatively strong in
relation to electrical coupling. Importantly, moreover, this
multiphasic rhythm unification occurs only if there is sufficient
heterogeneity of inhibitory or electrical coupling. The opposing action
of electrical and inhibitory coupling between two network cells is also
not a new phenomenon, especially in the stomatogastric system
(Graubard and Hartline 1987
). However, the
novelty of the present study has been to investigate the competition between these two types of coupling throughout a population of neurons
organized in multiple inhibitory networks operating at different
inherent cycle frequencies.
The other new effect of widespread electrical coupling reported
here, which accompanies the unification and eventual synchronization of
the population of neurons, is a gradual damping of individual neuronal
oscillations. With weak coupling, the damping of neuronal oscillations
results from the electrical interaction of individual cells with
surrounding networks, which generate low-amplitude NPs due to a still
asynchronous coordination within each network. By contrast, when cells
are synchronized by strong electrical coupling, the NPs generated are
of large amplitude, and there is no damping effect resulting from a
single-cell-NP interaction. In this state the damping is produced by
the synchrony itself. As they depolarize, individual cells will release
more inhibitory transmitter as well as receiving an increased amount of
inhibition from their synchronized neighbors. On the other hand, during
hyperpolarized phases they are weakly inhibited or not at all
(depending on the gradient of graded transmitter release as a function
of voltage). If sufficiently strong, this mutual inhibitory effect can
lead to a complete collapse of entire network activity.
The results described here have important implications for
communication between networks in which oscillatory neurons are interconnected by synaptic inhibition. Many such networks are responsible for rhythmic motor behavior (Marder and
Calabrese 1996
). A possible role of electrical
coupling in such systems could be to coordinate multiple network
function without synchronizing the activity of their constituent
elements. Moreover, since coupling efficacy is subject to ontogenetic
(Kandler and Katz 1995
; Pienado et al.
1993
; Walton and Navarrete 1991
) and/or
modulatory (Baldridge et al. 1998
; Hatton and
Yang 1996
; Johnson et al. 1993
) control, then a powerful mechanism for the dynamic reorganization of multiple network operation is potentially available.
Interestingly, the transition in network interactions seen in Fig. 9,
A and B, for example, strongly recalls
alterations in lobster STNS network function seen during the course of
development, where separate adult circuits emerge from a single
embryonic network that expresses small amplitude oscillations
(Casasnovas and Meyrand 1995
). Moreover, recent data
indicate that fundamental adult traits are already embedded in the
immature STNS but are masked by some developmental mechanism (Le
Feuvre et al. 1999
). Our present results suggest such a
mechanism: adult networks may be present in the same neuronal
population early in development but are forced to express a single
embryonic pattern by widespread electrical coupling.
A role for oscillation damping is still unclear. Indeed an initial
concern was that the relatively small amplitude of oscillation in
embryonic stomatogastric (STG) neurons simply resulted from cell damage due to microelectrode impalement. However, notwithstanding the fact that the size of these embryonic neurons, although
considerably smaller than their adult counterparts, are still larger
than most neurons that have been studied with intracellular recordings
in the vertebrate nervous system, there is evidence suggesting that major cell injury was not responsible for the small oscillations in the
STG neurons. First, comparison of intrasomatic activity with
intracellular recordings from target muscles before presynaptic motorneuron impalement confirmed that all network output parameters (cycle period, burst duration, spike frequency, and phase
relationships) remained unaltered. Second, soma impalement never led to
tonic firing, which would be expected to occur during any generalized damage-induced membrane depolarization.
A biological role for oscillation damping therefore is that it could
serve to match bioelectrical behavior with neuronal size by preventing
the electrotonic invasion of large-amplitude oscillations throughout
the entire structure of small embryonic neurons. Another related
possibility is that in a given network assemblage, the reduction of
oscillation amplitudes below spike threshold (see Fig. 8) may
immediately and reversibly prevent impulse-mediated communication with
other parts of the nervous system or the periphery, while any graded
transmitter signaling at central synapses within the population is
maintained. It is also important to realize that specific changes in
coupling strength (see Fig. 9C), such as a retention of
certain pathways during a developmental reduction in coupling
(Spitzer 1982
; Walsh et al. 1989
) or
their alteration by modulatory instruction in the adult nervous system
(Baldridge et al. 1998
; Hatton and Yang
1996
; Johnson et al. 1993
), may sustain multiple
network unification without causing damping of neuronal oscillations.
Finally, our data point to the possibility of considering neuronal
information transfer not only at the level of cell-to-cell communication, but also at the level of signaling between neuronal ensembles via synthetic representations of their activity conveyed through NPs.
For a model cell to become an endogenous oscillator it is
necessary that the fast current given by
This work was supported by the Polonium Program, Komitet Badan
Naukowych Status Grant 18/st, the Ministère de
l'Education Nationale de la Recherche et de la Technologie, and the
Conseil Régional d'Aquitaine.
Address for reprint requests: P. Meyrand, Laboratoire de Neurobiologie
des Réseaux, Université Bordeaux I and CNRS, UMR 5816, Avenue des Facultés, 33405 Talence, France (E-mail:
p.meyrand{at}lnr.u-bordeaux.fr).
Received 7 May 2001; accepted in final form 20 August 2001.