Biologie Cellulaire et Moléculaire du Neurone (Institut National de la Santé et de la Recherche Médicale U261), Institut Pasteur, 75724 Paris Cedex 15, France
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ABSTRACT |
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Faure, Philippe, Daniel Kaplan, and Henri Korn. Synaptic Efficacy and the Transmission of Complex Firing Patterns Between Neurons. J. Neurophysiol. 84: 3010-3025, 2000. In central neurons, the summation of inputs from presynaptic cells combined with the unreliability of synaptic transmission produces incessant variations of the membrane potential termed synaptic noise (SN). These fluctuations, which depend on both the unpredictable timing of afferent activities and quantal variations of postsynaptic potentials, have defied conventional analysis. We show here that, when applied to SN recorded from the Mauthner (M) cell of teleosts, a simple method of nonlinear analysis reveals previously undetected features of this signal including hidden periodic components. The phase relationship between these components is compatible with the notion that the temporal organization of events comprising this noise is deterministic rather than random and that it is generated by presynaptic interneurons behaving as coupled periodic oscillators. Furthermore a model of the presynaptic network shows how SN is shaped both by activities in incoming inputs and by the distribution of their synaptic weights expressed as mean quantal contents of the activated synapses. In confirmation we found experimentally that long-term tetanic potentiation (LTP), which selectively increases some of these synaptic weights, permits oscillating temporal patterns to be transmitted more effectively to the postsynaptic cell. Thus the probabilistic nature of transmitter release, which governs the strength of synapses, may be critical for the transfer of complex timing information within neuronal assemblies.
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INTRODUCTION |
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The nature of the neural code
has led to much speculation (for a review, see Buzsaki et al.
1994; Eggermont 1998
; Fuji et al.
1996
). For example it has been proposed (Hebb
1949
; Hopfield 1995
; Perkel and Bullock
1968
; Von der Malsburg 1981
) that the coding of
information in the CNS emerges from different firing patterns. Such
codes may include the rate of action potentials (Georgopoulos et
al. 1986
; Shadlen and Newsome 1998
),
well-defined synchronous activities of the "gamma" type (40 Hz)
particularly during binding (Singer 1993
), and more
complex temporal organization of firing in large networks
(Nicolelis et al. 1995
; Riehle et al.
1997
).
Relevant to the present study, it has been suggested that chaos, found
in several areas of the CNS (Pei and Moss 1996;
Schiff et al. 1994
), may also contribute to the neuronal
code (Skarda and Freeman 1987
, 1990
; So et al.
1998
; van Vreeswijk and Sompolinski 1996
). But
the validation of this hypothesis requires a demonstration that
deterministic patterns can be, and are effectively, transmitted along
neuronal chains.
This issue faces numerous difficulties, particularly in in vivo
preparations, due to the variability of the ongoing activity in neurons
called synaptic noise (SN) (Brock et al. 1952). This noise has been first attributed to a "random synaptic bombardment" of the recorded cells. Except for an early claim (Calvin and
Stevens 1967
) and some recent reports (Arieli et al.
1995
1996
), the view according to which SN
degrades neuronal functions has remained prevalent over the years (for
a review see Ferster 1996
). More important, this process has been
commonly assumed to be stochastic (Calvin and Stevens
1967
; Shadlen and Newsome 1998
;
Softky and Koch 1993
), and it has been most often
modeled as such (Mainen and Sejnowski 1995
;
Stevens and Zador 1998
). Therefore recent studies on
this intriguing phenomenon have mostly concentrated on whether or not,
and in which conditions, such a Poisson process contributes to the
variability of neuronal firing (Shadlen and Newsome 1994
,
1995
; Softky 1995
). However, the renewed
interest in SN leaves open the question of whether specific information about the state of firing of the presynaptic networks can be extracted from SN despite its random time appearance. The variability in both the
amplitude and time of occurrence of the synaptic responses that build
up SN precludes the sole use of conventional methods in solving this problem.
In this paper, we analyze the fine structure of SN in terms of timing
and amplitude. SN was recorded intracellularly from the Mauthner (M)
cell of teleosts, the command neuron of the adversive reaction to
external stimuli (Zottoli 1977). Specifically we want to
understand how SN reflects the state of the presynaptic networks and
how synaptic junctions are involved in the transmission of this
information. For this purpose, we examined physiological recordings
using analytical tools based on nonlinear dynamics (for a review, see
Abarbanel 1995
; Schreiber 1999
) already
successfully applied to decipher the complexity of other neuronal
systems (Guckenheimer and Rowat 1997
; Pei and
Moss 1996
; Schiff et al. 1994
). Our results indicate that, surprisingly, the fluctuating properties of synapses govern the degree to which complex activities in presynaptic networks are recapitulated postsynaptically and that this process is facilitated by a classical paradigm of learning.
Part of this work has been presented in an abstract form (Faure
and Korn 1998a).
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METHODS |
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Electrophysiological recordings
In the M cell of teleosts, the command neuron of the escape
reaction from adversive stimuli, SN is inhibitory and is generated by
two groups of glycinergic interneurons (Fig.
1A) one of which is driven by
auditory inputs (Faber and Korn 1978). This activity was
recorded in vivo in a quiet auditory environment with KCl-filled microelectrodes in the M-cell lateral dendrite of anesthetized adult
goldfish (Carassius auratus, n = 14) and
zebrafish (Brachydanio rerio, n = 16) as
described in Faure and Korn (1997)
. Because the
inhibitory postsynaptic potential (IPSP) in the M cell is hard to
detect as a potential change (Furshpan and Furukawa
1962
), Cl
was iontophoretically injected through the
recording microelectrode until large and stable full-sized depolarizing
collateral IPSPs evoked by antidromic activation of the M axon
(Faber and Korn 1982
) were recorded. Thus the IPSPs
comprising SN also appeared as depolarizing potentials (Fig.
1B, top). Recordings were digitized at 12 kHz and
filtered at 3 kHz with a low-pass Bessel filter.
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Data analysis
An approximation of the time derivative of the recorded signal
was calculated using first differences. The peak amplitudes and times
of occurrence of the resulting upstroke "spikes" constitute the
signal that was subjected to further analysis. These spikes could be
easily resolved against the background noise. The spike train was
sieved by ignoring all events whose amplitudes fell below a specified
threshold, . Thus the spike sequence submitted for further analysis
depends on both the signal itself and the imposed threshold.
The derived trains were examined with several techniques. Interspike
event histograms were constructed using a kernel-based density
estimator (Parzen 1962). The power spectrum of the spike point process was estimated using the discrete-time Fourier transform of the 12 kHz signal set equal to 1 at the time of events and 0 elsewhere.
Nonlinear structures in the spike trains were examined graphically
using return maps, also referred to as Poincaré maps (PMs) (Faure and Korn 1997, 1998b
; Garfinkel et al.
1992
). The PMs were constructed by scatter plotting each
interval between two successive events I(n + 1)
versus the previous one I(n). Note that in all maps presented in this report, each point corresponds to consecutive pairs of intervals (I1,
I2) among three IPSPs. That is for the first data point of the illustrated series,
I(n) = I1
and I(n + 1) = I2, whereas for the second one,
I(n) = I2,
I(n + 1) = I3, and so on.
Quantitative measurements of possible nonlinear determinism, or otherwise stated of the degree to which the studied signals can be distinguished from random processes, were made as explained in APPENDIX A.
Modeling
A mathematical representation of the presynaptic networks and
their dynamics consisting of four coupled model interneurons was built
following Hindmarsh and Rose (1984) and Rose and
Hindmarsh (1985)
. In this physiologically relevant and widely
used model (Abarbanel et al. 1996
; Hansel and
Sompolinsky 1992
; Keener and Sneyd 1998
), each
neuron i = 1, 2, 3, 4 is characterized by three time-dependent variables: xi, the
membrane potential; yi, a recovery variable; and zi, a slow adaptation
current. Let
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(1) |
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(2) |
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(3) |
Synapses linking neurons were formalized as
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(4) |
Simulations produced different network behaviors depending on the value
of gin = gin(t = ttp). These were synchronized antiphase bursts of action
potentials for strong coupling, i.e.,
gin 10, asynchronous and
intermittent firing for intermediate coupling 3.5
gin
10, and continuous firing of
the neurons with slight fluctuations around a mean frequency for small
coupling values such as gin
3.5.
Simulations for this paper corresponded to the third case.
Induction of LTP
LTP of M-cell inhibitory synapses was produced by trains of
sounds delivered by a loudspeaker placed in the vicinity of the fish
(50 ms, 500 Hz, and 75 dB) repeated every 4 s over 3 min (Oda et al. 1998). As in previous studies (Korn
et al. 1992
; Oda et al. 1995
), the method for
assessing the resulting increase of inhibitory synaptic strength was
based on measurements of the reduction in the antidromic spike height
due to the inhibitory shunt. Since this action potential propagates
passively into the soma (Furshpan and Furukawa 1962
),
any conductance change can be calculated as r' = (V/V')
1, where V and
V' are spike amplitudes in the absence and presence of
inhibition, respectively. This expression represents the ratio (or
fractional conductance)
GIPSP/Gm, the two terms being the inhibitory and resting conductances,
respectively (Faber and Korn 1982
).
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RESULTS |
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In previous studies, we have shown that nonrandom patterns can be
observed in SN (Faure and Korn 1997, 1998b
), but their
detailed organization has not been elucidated despite their striking
appearance in the form of well-delineated triangles in PMs. Thus
several questions have been addressed in the present study. The first was to identify the classes of dynamical systems and related synaptic events that can produce such geometric figures. The second was to
determine what can be inferred about the functional organization of the
presynaptic networks that generate the recorded SN. This led us to
investigate how the firing patterns of these networks are transferred
to postsynaptic cells. The overall conclusion of this investigation is
that even though synaptic transmission is probabilistic, SN is a true
signal that offers insight on the state of firing of the presynaptic networks.
Periodic components in SN
The recorded signal V(t) showed large oscillations made of successive inverted IPSPs with a magnitude up to several millivolts (Fig. 1B, top). The time derivative dV/dt of this signal provides an index of the onset and size of each event: the beginning of an IPSP was apparent as a rapid increase in slope reaching a maximum during the rising phase of the synaptic potential (Fig. 1B, bottom). The size of the resulting spike was proportional to that of its parent IPSP.
Figure 1C illustrates the basic method used for this study.
Subsets of events were selected, according to their amplitudes, by a
threshold in such a manner that as this threshold was lowered an
increasing number of events was included in the resulting time series.
The use of multiple-threshold levels to produce corresponding trains of
events derived from the same dV/dt recording
(Fig. 2A, 1 and
2) allowed us to take into consideration the information contained in both their timing and relative amplitudes.
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For example, the distribution of interspike intervals (ISI) obtained
with the events which crossed the highest (=
1) were multimodal (Fig. 2B1).
This confirmed their regular periodicity (see also Faure and
Korn 1997
), which can also be evidenced with autocorrelations
(Hatta and Korn 1999
). But for lower
, this periodicity was blurred, and time intervals possessed no obvious structure when the lower-amplitude IPSPs were taken into account (Fig.
2B, 2 and 3).
Power spectra of the spike point process showed additional organization
of the trains. In the illustrated case, even at
3 (Fig. 2C), there was a broad peak
between 60 and 80 Hz. This result was interesting but difficult to
interpret. The simplest explanation, that the events were approximately
periodic with a period of roughly 12-16 ms, was clearly ruled out by
the ISI histogram (Fig. 2B3), which shows that interspike
intervals (ISIs) were typically less than 10 ms. Another
interpretation, which as shown in the following text is correct, is
that this train comprised several interwoven periodic components.
Unfortunately, such a structure is difficult to deduce from the power
spectrum since there are few objective criteria to count the number of
peaks particularly when drifts in frequencies may confound the
situation. Furthermore power spectra are insensitive to phase
relationships and therefore cannot provide insights into the
relationship between oscillators that is typically a nonlinear phenomenon.
In contrast the PMs showed a highly structured pattern. At a
high-amplitude threshold, 1, the IPSPs were
strongly periodic. This appeared (Fig.
3A1) as a small circular
cloud: each ISI was followed by an interval of approximately the same
duration corresponding, in the illustrated experiment, to a principal
frequency, fp, the value of which (60 Hz) was the same as that of the
main peak in the ISI histogram of Fig. 2B, 1 and
2. At
2, more events were included
in the PM which showed a triangular (Faure and Korn
1997
) or, better stated, a signal-flag pattern. The summit of
this motif was centered on fp. This striking figure was observed in
70% (i.e., 21/30) of the experiments where SN could be recorded in
stable conditions. At this level there was also an outlined space
filled on the lower left triangle (Fig. 3A2), and at
3 there was only the lower left triangle (Fig.
3A3).
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A systematic search was carried out to determine if other periodic
components were buried in SN. For this purpose, events associated with
fp were subtracted from the time series isolated by
3 (Fig. 3B1), and the maps were
reconstructed with the remaining events. This procedure disclosed a
distinct second signal-flag pattern, centered on a secondary frequency
fs. In this experiment, fs was equal to 68 Hz (Fig. 3B2),
and overall we found fp < fs in the 12 of 21 experiments where
additional triangles could be revealed.
These results suggest that the IPSPs in SN are organized in a primary periodic train of high-amplitude and frequency (fp) and in a second train of somewhat lower amplitude and frequency (fs). As shown in the following text, other intervening and smaller events correspond to at least a third oscillator, (ft).
Interpretation of the PMs
A simple theoretical analysis consistent with experimental data
helps to interpret the signal-flag pattern. Consider a sequence of
events (Fig. 4A) consisting of
evenly spaced IPSPs (labeled P) having an amplitude greater than
P and period of
P and
another sequence of smaller events (labeled S) having an amplitude near
S and period
S, which
are intermingled with the larger P components. The sequences that are
used to construct the ISIs depend on the setting of the amplitude
threshold. Overall, maps constructed with interwoven periodic S and P
events and with some missing S will produce a PM with points scattered
on the four lines shown in Fig. 4B.
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Specifically, when the threshold is between P
and
S, only the P events will be detected. The
resulting sequence can be denoted PsPsPsPs, where capital letters stand
for events that are above the threshold. At this level, the sequence of
ISIs will be
P,
P,
P, ... and the PM will be a single dot at the position marked PsPsP. A
small amount of random variations in
P will
broaden this dot into the type of the circular cloud seen experimentally.
At a lower threshold, near S, some S events
will be detected and the resulting sequence of events is PSPSPsPS. The
ISI comes from consecutive triples, e.g., PSP, SPS, or SPP. Pairs of
intervals from PSP triples, e.g.,
I1,I2
and
I3,I4,
in Fig. 4A span the period
P, and
they appear in the PM along the diagonal line marked
P. For SPS triples, the dot can be anywhere in
the square bounded by
P, but when the S-type
events are periodic, SPS triples lie on the diagonal line corresponding
to
S. When S events fall below
S, triples will be of the form PsPS, SPsP, or
PsPsP. These appear on the vertical or horizontal lines of the
signal-flag pattern or at their intersection, respectively.
Finally, PMs constructed with thresholds below
S also exhibit points in the lower left
triangle already evident in Fig. 3A, 2 and 3.
These points correspond to the events denoted T that were also assumed
to be part of a periodic sequence.
A careful examination of the experimental PMs confirmed this
interpretation. For example, the principal and secondary periods P = 16.25 ms and
S = 14.4 ms detected at the summit of the triangles in Fig. 3,
A2 and 2, were also apparent at the border of
the highest density areas when the PMs were converted into density maps
(Fig. 4C). Applying the same protocol to the triangle
obtained after excluding P events disclosed a third period
T = 13.3 ms at the lower edge of the
signal-flag pattern (Fig. 4D). The values of these three
periods helped to make sense of some of the peaks revealed by the power
spectrum of Fig. 2C (arrows).
Are the presynaptic oscillators coupled?
The experimental PMs, corroborated by the power spectra,
indicated that activities in the presynaptic interneurons that generate SN are rhythmic. However the linear signal-flag structures in the
experimental PMs were broader than those expected if the P and S events
were exactly periodic (in which case they would lie exactly on the 4 lines of Fig. 4B). Two explanations were possible. One was
that P,
S, and
T were independent of each other and varied
randomly. The other was that, as in other neuronal systems (Keener and Glass 1984
; Pei and Moss
1996
), the parent oscillators of the P, S, and T events are
coupled to one another, thus producing fluctuations in the periods.
An analysis of the time intervals between the IPSPs comprising SN was carried out to distinguish between three distinct alternative possibilities. First, if the two oscillators P and S are independent, S events will occur with equal probability at any position in PP intervals (Fig. 5A, top). Second, if the oscillators are phase locked, for example by a strong synaptic path with a fixed delay, PS intervals will remain constant (Fig. 5A, middle). Finally, in case of a functionally weak synaptic coupling, the oscillators exert complex influences on each other and depending on the previous timing of P and S events, the phase of a subsequent one will be advanced, retarded, or remain the same (Fig. 5A, bottom).
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When PSPSP sequences corresponding to periods
P and
S were
extracted from SN and ranked as in Fig. 5B1, we found that
intervals P1P2
and S1S2 were
strongly correlated and that S events were not homogeneously
distributed between P ones as would be the case if the oscillators were
independent. This phase relationship was quantified, as illustrated by
the plot of Fig. 5B2 (black circles) in which the regression
line shows a statistically significant (P < 0.001)
rejection of the null hypothesis that the SS and PS intervals were
independent. Furthermore when the largest, though unclear (due to their
timing) S events were incorporated in the analysis (Fig.
5B2, blue circles), the slope became even more pronounced.
Thus the plot of Fig. 5B2 can be taken as indicating that
the S events did not occur during the first 20% of the PP cycle, SS
intervals were dramatically prolonged when a S event occurred in the
first half of the PP interval, and these intervals were shorter when
the S events occurred later in the PP sequence. These three forms of
mutual interaction are consistent with a weak coupling between oscillators.
This coupling was not strong enough to phase-lock the S and P events to
a stable and constant interval, but it could produce short sequences
that exhibited almost constant phase relationships between the P and S
oscillators. Advance and retreat patterns (ARPs), similar to those seen
previously (Faure and Korn 1997), were observed in the
PMs (Fig. 6A). These stemmed
from the slow drift in phase between S and P events. When the two
oscillators were half a cycle out of phase they appeared as a fixed
point (Fig. 6A, 2) but since they were not phase locked,
this point was unstable and the next points labeled 3 and 4 in the PM
diverged along a well-defined path resembling that of unstable periodic orbits (UPOs) often associated with chaos (So et al.
1998
). Other types of period 1, 2, and 3 orbits were also found
(Fig. 6B).
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These different configurations correspond to the various sequences of time intervals between events outlined in Fig. 6C. They stress the diversity of temporal structures contained in SN. The period-1 orbits correspond to a single interval between successive events. Period-2 orbits correspond to two different alternating intervals, while period-3 ones include three distinct sequential repeated intervals and these iterations can be generalized to n. The construction of the ARPs is more complex: the successive intervals converge toward and tend to stabilize around a fixed one, but they rapidly escape following distinct paths as illustrated in Fig. 6A.
Finally it should be stated that the measures of determinism, i.e., the
percentage of determinism (%det) and the µ() entropy (see details
in APPENDIX A), were statistically significant in 19 of the
21 experiments when compared with surrogates, confirming the nonlinear
properties of SN.
Complex patterns in presynaptic networks
Numerous reports have demonstrated that coupled neurons can behave
as oscillators and generate a vast repertoire of dynamic responses,
ranging from periodic to chaotic firing patterns (Abarbanel et
al. 1996; Borisyuk et al. 1995
; Hansel
and Sompolinsky 1992
; Rinzel et al. 1998
). Since
periodic events compatible with the involvement of coupled presynaptic
"oscillators" were detected in SN, we assumed that the role of the
M cell in the oscillations is simply a read-out function, and we
investigated whether coupling between inhibitory interneurons
terminating on the M cell can produce complex patterns similar to those
observed in actual data. We found that a deterministic model, which
does not involve random fluctuations, can reproduce all the major
features of the signal-flag geometry, including the broadening of the
diagonal PSP and SPS lines of Fig. 4.
Four interneurons were modeled as described in METHODS. They were linked by inhibitory synapses (Fig. 7A) generating IPSPs having a fixed latency and a constant amplitude to eliminate all sources of randomness. Values of this model's parameters were set to obtain low-frequency periodic patterns of firing of the same order than those of the different classes of IPSPs revealed in PMs.
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Figure 7B shows that each of the neurons produced trains of action potentials that were roughly periodic but fluctuated around a given mean period. Such deterministic fluctuations might account for the complexity of the experimental time series (see following text). However, the sum of the impulses produced by the four neurons took the undefined form of a random process and the corresponding return maps (Fig. 7C) were similar to those obtained by plotting events selected in actual data by a low threshold, as in Fig. 3A3.
Since most highly structured PMs were those constructed with
intermediate thresholds 2 and suggested the
interplay of rhythms from two oscillators, we focused our attention on
the behavior of two of the modeled neurons alone. For example, in the
case of Fig. 8A, the two
investigated cells fired with a mean frequency of 57 and 63 Hz,
respectively. Yet, the intervals between the action potentials in the
summed train were irregular with phase shifts analogous to those
produced by weakly coupled oscillators.
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Return maps constructed by plotting these intervals exhibited a
well-delineated motif made of series of points distributed around
P and
S (Fig.
8B1). That is, they were distributed in the same region of
the base of the signal-flag pattern obtained experimentally. One can
also note in the PM a few distant points (crossed arrow) that mark the
intervals between action potentials occurring before and after a pair
of synchronous ones (Fig. 8A, crossed arrow). They are the
precursors of the summit of a complete signal-flag pattern.
The scattering of points around P and
S indicates that the weak coupling between the
oscillators produced a deterministic dispersion of the time intervals
between events (Fig. 8B1), although the simulations did not
incorporate external sources of noise. These deterministic fluctuations
produced ARPs that resembled those observed in the experimental data
(not shown) and period-2 orbits (Fig. 8B2).
Role of synaptic properties in the transmission of presynaptic patterns
The various components of the signal-flag patterns were all
evident in the same train because some types of events were sometimes above and sometimes below threshold, the fluctuations of their amplitudes allowing the signal-flags to convey information about the
periodicities of each of the oscillators. In addition the distribution
of amplitudes allowed for gradual transitions among the types of
signal-flag patterns as the detection threshold was lowered with,
consequently, a reduction in the number of failures of detection.
Therefore by examining the fraction of missed events at each level of
, we could approximate the extent of the overlap of some IPSPs
produced by each oscillators (APPENDIX B1). The amplitude
fluctuations of these IPSPs were consistent with an involvement of
synaptic junctions. We found that indeed, connecting the formal
interneurons with terminal synapses that "released" transmitter in
quanta according to principles established at chemical junctions
allowed the model to reproduce the hierarchical features of the PMs.
Chemical transmission is governed by two parameters, n and
p, where n represents the population of basic "quantal
units" q capable of responding to a nerve impulse, and
p their average response probability (Del Castillo
and Katz 1952). Then the product np is the so-called
mean quantal content, which is equal to the average number of quanta
released by a given junction during successive trials, while the
product npq, where q is the size of a quantum (set to 1 in our simulations), determines the synaptic strength. Since
at central synapses transmitter release can follow a simple or a
compound binomial statistics, where p is the same or is
different for every site of release established by a neuron on its
target (for a review, see Korn and Faber 1991
;
Redman 1990
), we tested these two models in this study.
First, a simple binomial model, previously shown to account for the
amplitude fluctuations of IPSPs evoked in the M cell by its presynaptic
interneurons (Korn et al. 1982), was used with values
for the terms n and p in the range of those
derived in earlier experiments (Korn et al. 1986
).
When the neurons had the same quantal content np, all IPSPs
fluctuated in the same range (Fig.
9A, 1 and 2) and
the signal-flag pattern did not appear in the PM (Fig. 9A3).
This result was easily explained by the fact that any threshold
detected the same proportion of IPSPs regardless of their parent cell
and despite small changes at the extremities of the inverse cumulative
densities of the amplitude events (Fig. 9A2, ). These
slight discrepancies were due to small differences in the coefficient
of variation (CV) of the amplitude distributions of the IPSPs produced
by each cell. Similar conclusions were reached with other runs,
whatever the values attributed to n and p of each
interneuron, as long as the four modeled cells had an identical quantal
content (a justification of this rule, which also pertains for compound
binomial statistics, can be found in APPENDIX B2).
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When the model was modified to include different np products
that distinguish presynaptic neurons from each other (Korn et al. 1986), the signal-flag pattern was restored in the maps and the ARPs reappeared, as well as period-2 orbits (not shown). This result is illustrated in Fig. 9B, 1-3, for which the
"neurons" had a quantal content of 13.3, 6.2, 3.7, and 1.86, respectively. Again, this result can be easily explained. As shown by
Fig. 9B, 1 and 2, at intermediate values,
identified preferentially IPSPs produced by the first two cells. For
example, when
was set to select the same proportion of events
(i.e.,
42%) as in Fig. 9A2, 100% of events from
cell 1 and 60% from cell 2 were suprathreshold, against only 10 and 0% from cells 3 and 4.
Second, the model was implemented with terminals "releasing" according to nonuniform p's. As illustrated in Fig. 10, the results were almost indistinguishable from those described in the preceding text. That is, a signal-flag pattern, ARPs and period-2 orbits became only apparent in PMs when the np products were different. This finding was not surprising since the distribution histograms of IPSPs modeled by a compound binomial are roughly similar to those from a simple binomial distribution but with a smaller CV than with a simple binomial statistics.
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Finally it should be noted that in these simulations values of
n and p were inversely correlated as
experimentally observed at M cell's inhibitory connections
(Korn et al. 1986). However, this need not be the case
(see APPENDIX B2).
Experimental validation
To verify the involvement of synaptic efficacies, we conducted a
series of experiments, taking advantage that the strength of the M
cell's inhibitory synapses are modified in vivo by LTP, a classical
paradigm of learning that can be induced in teleosts by trains of
sounds emitted in the vicinity of the fish. This form of LTP is due to
an increase of the presynaptic parameter of release p, while
n and q remain unaffected (Oda et al.
1995). As in previous reports, the potentiation of the
inhibitory synapses was quantified using a parameter r', or
fractional conductance, which defines the degree of the inhibitory
shunt (see METHODS).
As expected the amplitude of IPSPs comprising SN was increased during
LTP (Fig. 11, A1 and
B1). Furthermore return maps constructed with a
high-threshold (2) were markedly different
before and after the conditioning sound trains (Fig. 11, A2
and B2). That is, LTP strengthened the two-oscillator
triangular pattern in the PM, as further evidenced by comparing the PMs
constructed with a lower threshold,
'2 (Fig. 11,
B2 and C) and by the 8% increase of the %det
(Fig. 11D). On the other hand
P and
s remained the same (Fig.
12), suggesting a stability of the
dynamics in the network.
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In four experiments, r' was increased by 29 ± 8.1%
(mean ± SE) after the learning protocol, indicating LTP of the
commissural synapses (Korn et al. 1992), and the %det
was enhanced by 14.3 ± 2.1%. Adding three other experiments,
during which LTP was only assessed by comparing the amplitude of IPSPs
(to avoid further modifications of SN by control sounds), this value
became 11.3 ± 1.8%. In another cell, the sounds produced no LTP
and the determinism was unchanged.
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DISCUSSION |
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Taken together these results suggest that the information contained in SN permits the dynamics of the presynaptic networks to be reconstructed. This information is contained both in the intervals between IPSPs and in their amplitudes.
Our data can be generalized as follows. Several combinations of P and S events in triplets, some of which result from failures to detect events, are necessary to construct a complete signal-flag pattern. This prerequisite is guaranteed by the "separating power" of the largest np products at the terminals of neurons that generate the SN. On the other hand identified dynamical sequences (ARPs, period-n orbits and other signs of determinism) are found at the base of the signal-flag pattern, and the clarity with which they appear is related to the emergence of both P and S events in the time series. This second condition evidently imposes an irreducible separation between the np products (see APPENDIX B2) that allows the presynaptic dynamics to become predominant in the SN.
Validity of the coupling hypothesis
Two important issues had to be considered before accepting that the coupling between presynaptic interneurons accounts for the patterns observed in SN.
First, might we have incorrectly classified as P, S, or T activities?
Our methodology for displaying the coupling between the P and S
oscillators relies on the detection of events and their classification
as P, S, and T. Thus any failure of detection or classification that is
related to the timing of the spikes could artifactually influence the
display of coupling. However, when constructing Fig. 5B2, we
remained conservative and considered only events (shown in black)
belonging to unambiguous PSPS quartets. P, S, and T events were
selected using a criterion that combines amplitude with timing
information. Specifically, P events were obvious and, despite possible
confusions between S and T events due to overlap of their size, most S
ones could be identified as such in Poincaré maps since they
appeared on the diagonal line corresponding to the S period. Second,
how reliably could we detect closely spaced events? Events closer than
a given time separation were not distinguished in the time series. This
issue was important because a sufficiently long lockout can produce the
illusion of coupling. But this lockout was approximately 0.5 ms, and
there are few points with short in Fig. 5B2 simply
because there were few PSPS quartets. Theoretical analysis shows that the 0.5 ms lockout is too short to produce artifacts mimicking the
pattern in Fig. 5B2. In confirmation, simulations with
uncoupled S and P oscillators show virtually no coupling even with
lockouts as large as 3.5 ms.
Two additional arguments reinforced the hypothesis of coupling. Data
points in the PMs were not uniformly distributed as they would be if
the variability of each frequency was random and, as demonstrated by
combined electrophysiological and histological studies (Korn et
al. 1990; Triller and Korn 1981
), commissural inhibitory interneurons presynaptic to the M cell are linked by chemical inhibitory synapses.
Other simulations can produce triangular maps. Such is the case for models based on one oscillator producing large events intercalated with smaller ones occurring at random (not shown) or on several independent and noisy oscillators, each making events of different amplitudes, and a fixed level of detection. However when examined in detail, such maps do not exhibit the fine temporal structure (periodic orbits, phase relationships) found in the SN recorded from the M cell.
Finally could nonlinearity in synaptic transmission
and/or the responsiveness of the M cell to its inputs create the
observed patterns? While this is conceivable, the most obvious sources of nonlinearity so far identified in the M cell system are unlikely to
be involved. In particular the effects of presynaptic depression that
reduces IPSP amplitudes at high rates of presynaptic firing are
stabilized at frequencies more than 33 Hz (Korn et al.
1984). In confirmation no correlation was found between the
amplitudes of the successive P events or between their amplitudes and
timing. At the postsynaptic level, we investigated whether the
nonlinear summation of potential change (Martin 1955
)
and, more importantly, the voltage dependence of the decay of IPSPs
(Faber and Korn 1987
) did affect the size of overlapping
events. Such seemed not the case since the amplitude distribution of S
events was the same during the first half of the PP cycle (which
includes the falling phase of the P events) and during the second one.
Neuronal correlates of the presynaptic oscillators
Nonlinear deterministic components have been identified in SN
(Faure and Korn 1997, 1998b
), and it is well established
that coupled oscillators can generate similar patterns, including
chaotic ones (Glass and Mackey 1988
). The frequencies of
the oscillators unmasked in this study are consistent with those of the
periodic components already noted in the M cell's inhibitory SN
(Faure and Korn 1997
; Hatta and Korn
1999
), which were in the range of the so-called gamma rhythm
observed in higher vertebrates (Jefferys et al. 1996
;
Singer 1993
).
Anatomical and physiological studies have shown that there are
more than four interneurons in the M-cell presynaptic network with the
estimates being in the range of at least 50 (Faber and Korn
1978; Korn and Faber 1990
). Thus it is
surprising that the contribution of a few oscillators can be
distinguished or, in other terms, that the amplitude of the IPSPs
produced by each of these oscillators seems to be ordered according to
their origin. Calculations of the average size of the P and S IPSPs may
help to address this issue. In the goldfish M-cell system, the size of
an inhibitory quantum is approximately 1% that of the full-sized collateral IPSP evoked by antidromic activation of the Mauthner cell
axon (Korn et al. 1982
). The mean unitary IPSPs produced by stimulations at 1 Hz of a single presynaptic interneuron comprise 5.8 quanta (Korn et al. 1986
). Furthermore their
amplitudes decrease in a known way at increased stimulating frequencies
due to presynaptic depression (Korn et al. 1984
). Based
on these values, estimations made in time series from three goldfish
indicated that the P and S IPSPs were, respectively, six to seven and
four to five times bigger than the average unitary response.
Several hypotheses that have not been tested in this study might
explain this amplitude distribution. One is that these IPSPs are
produced by the firing of a special set of cells, that of the rather
exceptional "superinterneurons," which evoke IPSPs of unusually
high amplitudes (Korn et al. 1986). The other is that a
number of presynaptic cells might fire simultaneously. Two signaling
mechanisms already demonstrated in the organization of the M cell's
inhibitory network could be involved: one is the chemical coupling
between inhibitory interneurons that has been shown to underlie
synchronization in theoretical models (Jefferys et al.
1996
; Traub et al. 1996
; White et al.
1998
; Whittington et al. 1995
) and in
experimental material (Bragin et al. 1995
; Whittington et al. 1995
). The second is that the somata
of the commissural interneurons are linked in teleosts via gap
junctions (Korn et al. 1977
), which favor their
simultaneous discharge including those of functionally related
inhibitory cells (Gibson et al. 1999
).
Arguments offered in APPENDIX B3 indicate that our results do not depend on whether the quanta are issued by synchronously firing cells or are all from a single presynaptic neuron.
Synaptic properties and the transmission of deterministic patterns
Analysis of the PMs indicated that the signal-flag patterns were
clearly delineated with maximum values of the measures of determinism
[%det and µ()] and the largest number of ARPs, when 99.3 ± 0.32% (n = 10), 49.7 ± 2.57%
(n = 10), and 12.1 ± 1.89% (n = 10) of P, S, and T events were included in the maps, respectively. These values agree closely with those of the np products
required for the construction of "meaningful" maps with models.
We observed in experimental data that the same sequence of event
intervals can produce qualitatively different patterns depending on the
level of the threshold used to window the sequence. Indeed, the
probabilistic nature of transmitter release means that there is a
graduated transition between the different patterns as is varied.
The notion that the probabilistic component p is critical here was confirmed by simulations in which the oscillators had different n's but with p = 1 at all
terminals, reducing the variance to
2 = 0. This resulted for any level of
in maps lacking signal-flag patterns. The nonzero variance that results from 0 < p < 0.1 guarantees that multiple components of the
patterns shown in Fig. 4B can be present in the signal
flags. The value of p sets the relative mixture of these
patterns. Changing p, as in LTP, modifies this mixture.
In this context, the probabilistic aspect of neurotransmission, whose
function in neuronal communication has not been clarified, becomes an
advantage rather than a limiting factor (Zador 1998) since it allows synaptic strengths to be adjusted in a manner that
shapes the transmitted information without needing to modify the
dynamics of the presynaptic network.
The emergence of the deterministic structures in the postsynaptic cell
with multiple inputs is made possible by the nonuniform values of
synaptic weights and the stochastic release of quanta. Furthermore the
transmission of qualitatively different patterns in the M cell may
allow encoding of a small repertoire of motor reactions as suggested by
the results of LTP which is known to underlie modifications of the
goldfish escape behavior (Oda et al. 1998).
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APPENDIX A |
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A deterministic system is one whose complete behavior can be expressed with an infinite precision in a mathematical description (usually a differential equation). Then if the initial values of all variables are known completely, the system's entire future history is predictable and can be calculated exactly.
A chaotic system, which often appears at first sight to be random, is also deterministic, but it displays what is called sensitive dependence to initial conditions. That is, solutions obtained with two different starting points can be profoundly different: if the initial conditions are infinitesimally displaced from each other, then the solutions diverge exponentially. The extent to which they are unpredictable depends on this rate of divergence.
The dynamics of deterministic systems are different in principle from random processes where prediction is impossible except statistically.
While the presence or absence of a specific form of determinism was not
the focus of this report (for this aspect of the work, see Faure
and Korn 1997, 1998b
), that of SN was quantified for two
reasons. The first was to confirm that SN could be distinguished from a
random process. The second was to estimate the modifications produced
by external sensory stimuli. Two parameters were used. Both detect
so-called recurrent patterns, i.e., sequences which approximatively
repeat themselves over time (hidden rhythms). They can be computed in
recurrent plots (Eckmann et al. 1987
), which are
particularly well suited for studies of biological processes.
Let x(i) be the ith point on the orbit
describing a dynamical system in a d-dimensional space, for
i = 1, ... , N. The recurrence plot (RP)
is an array of dots in a N × N square,
where a recurrent point is placed at the (i, j)
coordinates whenever the embedded vectors (xi,
xj), defined here by xi = [I(i), I(i + 1)] are
within a predetermined cutoff distance (). The organization of
recurrent points into diagonal line segments of length L
indicates the parts of the trajectories which are close (under
resolution
) during L successive time steps.
The first parameter was the percentage of determinism (%det)
(Weber and Zbilut 1994), which is the number of dots
included in diagonal line segments divided by the total number of dots in the RP. The second parameter was the slope, µ(
), of the
exponential decay of the histogram of the number of segments of length
L. As demonstrated elsewhere (Faure and Korn
1998b
), at limits, this slope is an estimation of the
Kolmogorov-Sinai entropy.
To confirm the nonlinear properties of the PMs, we constructed
surrogate data. A surrogate is an artificial set constructed from the
original data with constrained statistical properties that depend on
the null hypothesis being tested. The statistical significance of the
two parameters was examined with the null hypothesis that all forms of
determinism found in SN were brought about by the linear properties
(amplitude and frequency distributions) of the signal. For this
purpose, surrogates of the time derivative of the membrane potential
and of the raw spike trains, which matched both the amplitudes of the
signals and their power spectrum, were constructed using two distinct
methods despite their possible limitations when applied to time
interval series (Schreiber and Schmitz 2000). These were
the amplitude-adjusted method (Rapp et al. 1993
;
Theiler et al. 1992
) and the iterative surrogate technique (Schreiber and Schmitz 1996
), which involves
an iterative refinement of the latter.
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APPENDIX B |
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This appendix is concerned with the synaptic properties that can lead to the the signal-flag patterns. We treat the amplitude of each event (i.e., of an IPSP in SN) as random and seek to characterize the amplitude distributions of the P, S, and T events.
Appendix B1: percentages of detection
We need some statement about amplitude distributions of each
class of IPSPs to express their degree of overlap. Since the release
mechanism may obey simple or compound binomial or possibly other
statistics (if the quanta are not released independently), we represent
the probability distribution of the event amplitudes as a gaussian
distribution described by a mean amplitude m and a variance
2. We denote the mean amplitude of the
different types of events as mP,
mS,
mT and their variances by
P2,
S2,
T2.
As is conventional for a gaussian distribution, any percentile can be
depicted in terms of the standardized z measure. For example
the 2.5th percentile of a normal distribution lies at m + z0.025. As is found in standard
tables of the Gaussian distribution, z0.025 =
1.96.
The PMs give information about the fraction of events of each type
larger than a selected , that is, about the percentiles of the
distribution. So, the number of missed P's could be assessed in Fig.
3A1 by computing that of the overlong intervals (with In or
In+1
P),
which do not appear on the PMs due to the chosen scales. Since
P was known, one could interpolate the
position of the missing P's. This procedure indicated 52 of them
inferred among the 566, i.e., 92%, detected ones.
Generally in our experiments, there was a threshold at which 95% of P
and no more than 5% of T events were detected. This observation can be
translated into the z notation as:
mP + z0.05P >
and mT + z0.95
T <
(where z0.05 is negative). Combining the two statements we have
![]() |
(B1) |
Similar relationships can be found by comparing other classes of
events: the fact that approximately 50% of S and 2% of T are greater
than = 3,500 for the data of Fig. 3 indicates that mS + z0.50
S > mT + z0.98
T or,
substituting in the tabulated values for
z0.50 and
z0.98,
mS > mT + 2
T.
Appendix B2: implications of np values on the percentage of detection
m and 2 can be related to
the release parameters n and p. For each type of
events and assuming binomial statistics, we have m = np and
=
.
Relationships of the form of Eq. B1 can be used
to determine theoretical limits on the number of quanta potentially
involved in each type of events. For any nonzero value of
P and
T, this equation directly implies that mP > mT and therefore that
nPpP > nTpT.
In other words, there can be no signal-flag pattern if the
np products are equal. This rule can be generalized to a
compound binomial. If the n quanta are released with
different probabilities, then mP =
k=1nP
pk,P, mT =
k=1nT
pk,T,
P =
,
and
T =
.
Then it is easily shown that for fixed n and
m the compound binomial has always a smaller
than the
simple binomial distribution.
For both binomial-type models, the smaller are the values
P and
T, the closer
can be mP and
mT. In the theoretical limits where
P = 0 and
T = 0, it
is required from Eq. B1 that
nP > nT.
Using the coefficient of variation CV = /m, we have
from Eq. B1 the relation
![]() |
(B2) |
The CV also constrains the relationship between n and
p
![]() |
(B3) |
Appendix B3: extension to synchronized neurons
Considering N cells firing synchronously, where
cell i = 1, ... , N has
ni potential quanta for release, the total
number n of possible quanta is
![]() |
(B4) |
![]() |
(B5) |
The variance in the number of quanta released in the N cells
is
![]() |
(B6) |
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ACKNOWLEDGMENTS |
---|
We thank D. Ruelle (Institut des Hautes Etudes Scientifiques), R. Miles (Institut Pasteur), D. S. Faber (Albert Einstein College of Medicine, NY), P. Rapp (Medical College of Pennsylvania), and K. Hatta for experimental support.
This work was supported by Office of Naval Research Grant N00014-97-0226.
Present address of D. Kaplan: Dept. of Mathematics, Macalester College, St. Paul, MN 55105.
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FOOTNOTES |
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Address for reprint requests: H. Korn, Biologie Cellulaire et Moléculaire du Neurone (INSERM U261), Institut Pasteur, 25 rue du Docteur Roux, 75724 Paris Cedex 15, France (E-mail: hkorn{at}pasteur.fr).
Received 15 February 2000; accepted in final form 25 July 2000.
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REFERENCES |
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