1Institute for Nonlinear Science, 2Department of Physics, and Marine Physical Laboratory, Scripps Institution of Oceanography, University of California, San Diego, California 92093-0402; and 3Balaton Limnological Research Institute of the Hungarian Academy of Sciences, H-8237 Tihany, Hungary
![]() |
ABSTRACT |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Sz
![]() |
INTRODUCTION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The voltage behavior of
bursting neurons typically arises from the interplay of many ionic
currents, synaptic and intracellular processes (Calabrese
1998; Canavier et al. 1991
; Stein et al. 1997
). The neurons can cooperate to generate coordinated,
synchronized activity as a result of their synaptic coupling or in
response to external driving inputs. To investigate these properties
further, we have studied the responses to sinusoidal current injection of a small circuit of motor pattern generating neurons, both as individuals and as an electrically coupled group. Using linear and
nonlinear analytical techniques, we uncovered rich dynamical behavior
that was not analyzed in previous studies of entrainment (Ayers
and Selverston 1979
).
Periodic stimulation of nonlinear biological and physical systems
offers a natural method for investigating their intrinsic dynamics.
Stimulation by sinusoidal waveforms of the form A
sin (2ft) with different amplitudes A and
frequencies f can drive the system into a wide repertoire of
responses including periodic or chaotic temporal patterns
(Tomita 1986
). Periodic current injection has been used
to map the dynamical behavior of squid giant axons (Aihara
1984
; Kaplan et al. 1996
), snail neurons
(Chillemi et al. 1997
), and embryonic chick heart cells
(Glass et al. 1986
), as well as mathematical models of
neurons (Matsugu et al. 1998
; Wang et al.
2000
). Frequency- and intensity-dependent responses characteristic of deterministic nonlinear oscillators have been demonstrated in these experiments and models. Periodic stimulation of
nerve fibers or single neurons has also proven to be an effective tool
for examining the information processing strategies of the nervous
system at cellular level (Hooper 1998
; Segundo et
al. 1998
). Strong evidence of chaotic activity has been found
in hippocampal CA3 neurons receiving periodic inputs via mossy fiber
stimulation (Hayashi and Ishizuka 1995
). This indicates
deterministic dynamical behavior in complex multicomponent neural assemblies.
Previous work on sinusoidal stimulation of biological neural networks
has focused either on single neurons or very large populations of
interconnected neurons. In this paper we examine the response to
periodic forcing of a small subset of the very well characterized pyloric central pattern generator (CPG) network of lobster. Our experimental preparation, the pyloric CPG in the stomatogastric ganglion (STG) of the lobster, has served as an experimental model of
cellular plasticity and dynamic network organization
(Harris-Warrick et al. 1992; Marder 1997
;
Selverston and Moulins 1987
). The STG contains three
electrically coupled neurons that form the pacemaker group of the
pyloric CPG. These neurons [the anterior burster (AB) and 2 pyloric
dilators (PD)] receive inhibitory chemical synapses from other pyloric
motor neurons and generate a regular oscillatory pattern with a cycle
period of
1 s. The ventricular dilator neuron (VD) is weakly
connected to the PD and AB neurons by a rectifying electrotonic
synapse (Johnson et al. 1993
), but it is not commonly
considered as part of the pacemaker group. We show in Fig.
1 the circuit layout for the pyloric CPG
with the pacemaker group highlighted. The pyloric neurons also receive tonic neuromodulatory inputs from anterior centers of the
stomatogastric nervous system.
|
Using pharmacological tools, one can isolate the pyloric pacemaker
group from fast synaptic inputs coming from other pyloric neurons (Fig.
1B). This small functional group of neurons still produces
the main pyloric rhythm, often more regular than in intact preparations. This rhythm, however, can be observed only if
neuromodulatory inputs from anterior ganglia are left intact
(Bal et al. 1988). Besides the pyloric CPG, the
stomatogastric nervous system contains other motor pattern generating
networks, which typically run at a lower frequency. The interaction of
the pyloric circuit with the gastric network has already been studied
(Bartos and Nusbaum 1997
; Clemens et al.
1998
). The periodic forcing of the pacemaker neurons at low
frequencies offers a way to simulate, in some respects, the effects of
this intercircuit interaction.
The periodic forcing of oscillators is a familiar topic in physical
systems. The behavior of such oscillators is described in the parameter
plane with coordinates (A, f) (amplitude
and frequency of the forcing) (Glass and Mackey 1988).
In this plane different regions of synchronization form domains
narrowing to a point on the A = 0 axis; these regions
are known as "Arnol'd tongues." In our experiments the overall
oscillator is a composite of several nonlinear, complex oscillators;
however, the pacemaker group displays zones of synchronization and
Arnol'd tongues at frequencies near the harmonics and subharmonics of
the intrinsic pyloric frequency.
In addition to neurophysiological experiments, we performed sinusoidal
forcing experiments with an analog electronic implementation of the
modified Hindmarsh-Rose model of bursting neurons (Hindmarsh and
Rose 1984; Pinto et al. 2000
). The electronic
model neuron (EN) was based on nonlinear analysis of membrane voltage
time series, which show that only three to five degrees of freedom are
operating in the membrane voltage activity of the pacemaker neurons. We
have compared the EN's response to periodic current forcing with that
in living neurons, and we show them to be substantially the same.
![]() |
METHODS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
We performed experiments on adult intermolt California spiny lobsters Panulirus interruptus. The animals were obtained from a local fisherman and kept in large tanks of running, aerated seawater.
Preparation
The preparation has been described previously (Mulloney
and Selverston 1974). Briefly, the complete stomatogastric
nervous system containing the STG, the esophageal ganglion (OG), and
the two commissural ganglia (COG) was separated from the stomach and pinned in a silicone elastomer (Sylgard)-lined Petri dish. Dissection was performed in Panulirus physiological saline containing
(in mM/l) 483 NaCl, 12.7 KCl, 13.7 CaCl2, 10 MgSO4, 4 NaSO4, 5 HEPES, and 5 TES; pH was set to 7.40. The STG was desheathed using sharp forceps to facilitate access to the somata of the cells. The
stomatogastric ganglion was enclosed in a small petroleum jelly
(Vaseline) well for separate perfusion. The STG was superfused with
saline containing 7.5 µM picrotoxin (PTX) for a minimum of 30 min.
This treatment effectively blocked glutamatergic synaptic inputs to the
cells resulting in near complete pharmacological isolation of the
pyloric pacemaker group (Bidaut 1980
; Marder and
Eisen 1984
) (Fig. 1B). Anterior centers of the
nervous system (COGs, OG, and interconnecting nerves) were always
bathed in normal saline. The temperature of the bathing solution was
held at 16-18°C using a thermoelectric Peltier-device attached to
the bottom surface of the preparation chamber. Reduced configurations
of the pacemaker group were obtained by photoinactivating selected
neurons by filling them with 5,6-carboxyfluorescein and exposing
them to bright blue light (Selverston and Miller 1980
).
Electrophysiology
Membrane potentials of the neurons were recorded using glass
microelectrodes of 10-15 M filled with 3 M K-acetate plus 0.1 M KCl
solution. Membrane potentials of the cells were measured with
Neuroprobe 1600 current-clamp amplifiers (AM-Systems). Sinusoidal current command signals were generated either by a Tektronix CFG250 waveform generator or by computer running LabView 5.0 software and
equipped with a National Instruments PCI-MIO-16E4 AD/DA converter. The
amplitude of the sinusoidal current was between 0.5 and 2.5 nA, while
the frequency ranged from 0.2 to 4.0 Hz. The current was injected into
one of the neurons while recording from both the injected cell and the
cells coupled to it. The extra- and intracellular signals were acquired
at a 5- to 10-kS/s rate using the Axoscope 7 program (Axon Instruments)
running on a Pentium-266 computer.
Care was taken to monitor the stationarity of the intrinsic pyloric
bursting during the whole duration of the recording. We excluded data
from our analysis when the pyloric firing patterns were occasionally
disrupted by cardiac sac events or other potent synaptic inputs
arriving from other extraganglionic sources. The schedule of the
experiments was as follows: we recorded the control activity of the
neurons for 40-60 s, then applied the stimuli in separated 80- to
120-s trials allowing the cells to generate at least 100 bursts
(1,000 spikes) per stimulation. The frequency and amplitude of the
waveform was held constant during a trial.
Data analysis
Raw membrane potential data were visually inspected, while
detailed quantitative analysis was performed using series of spike arrival times. Establishing the arrival times of action potential occurrences and constructing the spike train
{t
To characterize the spontaneous and stimulus-induced firing patterns, a
number of functions and distributions were calculated from the spike
time data. The spike density function (SDF) was used as a
sensitive measure of the ongoing neuronal activity (Richmond et
al. 1987; Sz
). The SDF
provides a continuous estimate of the instantaneous firing rate and
allows one to detect even slight modulations of the firing pattern
(Paulin 1992
). The SDF is calculated by
convolving the time of each spike with a Gaussian-function K(t), called a kernel
![]() |
|
Pyloric pacemaker neurons of the intact network exhibit short, periodic
bursts of action potentials separated by silent periods of
hyperpolarization. Hence bursts can be considered as the basic events
of neural activity in the pyloric CPG. To characterize the timing of
bursts, we located the onset of each burst and generated a subsequent
discrete time series containing burst arrival times {t t
). As "proxy" coordinates for the
state space of those dynamics, we may use the values of BCP(i) themselves. The process is assumed to
occur in a d-dimensional space with vector coordinates
{BCP(i), BCP(i + k),
BCP(i + 2k), ... ,
BCP[I + (d
1)k]} at "time" i. The time
delay k is often taken to be unity, and while the dimension
d may be larger than 2, for display purposes d =
2 is often selected. Data vectors such as this provide a state space in
which many of the important dynamical properties of the underlying
dynamics are retained. We created d = 2 state spaces (also
called return maps) for the BCP(i) time series by
plotting points [BCP(i), BCP(i
+ 1)] in a two-dimensional plane. Equilibrium or stable fixed
points are seen as points in this plane along the 45° line. Period
two orbits of the dynamics are seen as pairs of points in the plane,
and more complex trajectories, higher order periodic orbits, or chaotic behavior produce more complex patterns. For comparison, we also constructed sequences and return maps of the interspike intervals: [ISI(i), ISI(i + 1)],
ISI(i) = t
t
Electronic model neuron
The electronic model neuron (EN) is an analog circuit
implementation of a four-dimensional neuron model using ordinary
differential equations with vector fields taken to be polynomials in
the dynamical variables [x(t),
y(t), z(t),
w(t)]. The model takes the form
![]() |
![]() |
![]() |
![]() |
(1) |
![]() |
RESULTS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Spontaneous firing patterns of neurons in the pyloric pacemaker group
Neurons of the intact pacemaker group (AB and 2 PDs) exhibited a
regular oscillatory firing pattern (Fig. 2A) when the fast glutamatergic inhibition was blocked (with PTX in saline). The pacemaker neurons were therefore synaptically isolated from other pyloric neurons, but still continuously receiving neuromodulatory inputs from anterior ganglia through the main stomatogastric nerve. The
bursting frequency was 1.6 ± 0.3 Hz (mean ± SD,
n = 25). The two PD neurons and the AB neuron began and
ended their bursts in synchrony, although individual spikes were not
precisely synchronized (Elson et al. 1998). The
monotonic bursting pattern was occasionally interrupted by intense
activation of synaptic inputs associated with the cardiac sac rhythm.
The frequency of these cardiac sac events tended to decrease during the
recording sessions, thus allowing us to obtain long stationary sections
of pyloric activity. Furthermore, as we show below, a low-frequency
modulation of the pyloric oscillation was also routinely observed, as a
consequence of interactions between the pyloric pacemaker neurons and
the slower gastric (and/or esophageal) network.
The spike density function of the regularly bursting PD neuron is shown in Fig. 2B. The SDF consists of separate equidistant peaks of similar amplitude and shape reflecting the periodicity and precision of the repetitive firing pattern. The Fourier transform of a longer section of the SDF is shown in Fig. 2D (linear amplitude scale). A large peak at 1.75 Hz indicates the frequency of the intrinsic pyloric bursting rhythm fib; the second harmonic at 2fib = 3.5 Hz appears as a minor peak. This is a typical example of Fourier spectra obtained from pacemaker neurons in stable pyloric oscillation.
The interspike interval (ISI) return map of the PD neuron in
the intact pacemaker group contains three separated clusters, which is
a clear consequence of the repetitive bursting (Fig. 2C).
Short ISIs associated with intraburst spikes are followed by
a long ISI associated with the relatively longer interburst duration. A compact V-shaped cluster (at short intervals, 0.025 s)
indicates precise and repetitive spiking in the successive bursts of
the PD neuron.
Photoinactivation of the main pacemaker cell AB led to disruption of this regular activity and irregular spiking-bursting patterns in both the PD and other postsynaptic pyloric neurons (n = 11). Immediately after photoinactivation of AB, the PD expressed irregular spiking behavior that evolved into more regular bursting activity over a time scale of hours. The membrane potential of the irregularly bursting PD is shown in Fig. 3A, with the corresponding SDF in Fig. 3B. The Fourier transform of the SDF (Fig. 3D) is now predominantly broadband. As the firing pattern of these PD neurons tended to be more "bursty" over a longer time scale, the resulting Fourier spectra became more sharp and peaked. However, the precise periodic bursting seen in the intact pacemaker group was never restored.
|
In the low-frequency range of the spectrum, a single peak appears at
0.2 Hz, indicating some remaining periodicity in the spike train of the
PD. This low-frequency rhythm was found in the majority of our
preparations, and it is considered to be a synaptic modulatory effect
from the gastric and/or esophageal network (Clemens et al.
1998; Russell and Hartline 1982
).
ISI return maps of the PD neuron before (Fig. 2C) and after killing AB (Fig. 3C) show major differences. In the latter case no three-clustered structure is present, rather a scattered cloud of points appears. Here, we observe a mixture of short and long ISIs with no well-defined clusters.
The firing pattern of the isolated AB neuron is shown in Fig.
4A. Here both PD neurons were
killed. The membrane potential trace of AB is similar to that of the PD
in the intact pacemaker group but with smaller spikes. The AB retained
its periodic bursting after photoinactivating the coupled PD neurons
(n = 4), showing its fundamental role as a pacemaker in
the pyloric rhythm. The frequency of the oscillation increased to
2.6 Hz from the normal fib
1.8 Hz
as shown by the Fourier-spectrum (Fig. 4D). A small peak
indicating the modulatory influence of the gastric rhythm is present at
0.3 Hz as well as accompanying peaks on each side of the fundamental
peak. These appear as the sum and difference of the pyloric and gastric
frequency (fib ± fgastric). The ISI return map
has three separate clusters (Fig. 4C) as that in the intact
pacemaker group.
|
Effects of periodic forcing on various configurations of the pacemaker group
Sinusoidal currents
Isin(t) = A
sin (2ft) with A of order 0.5-2.5 nA had
marked effects on the firing patterns and burst timing in all the
neurons tested. The effect of the forcing current was most evident and
clear on the neuron being injected. The other electrically coupled
neurons followed the induced rhythm, but usually with smaller amplitude
ratio at the forcing frequency.
Figure 2E shows the effect of
Isin(t) with A = 1 nA and f = 1.2 Hz on the bursting activity of the
PD neuron in intact pacemaker group. The interaction between the
periodic forcing and the intrinsic pyloric oscillation resulted in a
wide repertoire of burst patterns [n = 174 (8); number
of trials (preparations)]. Comparing the Fourier spectra of
spontaneously active (Fig. 2D) and forced (Fig. 2H) PD neurons, we find both similarities and differences.
In this example, f was 1.2 Hz, less than
fib. The original peak indicating the
frequency of the intrinsic bursting frequency
fib 1.9 Hz is still evident, but
additional peaks appear marking the frequency of the forcing current
(1.2 Hz, asterisk) and its harmonics. Furthermore, linear combinations
of the intrinsic and forcing frequencies appear as minor peaks (e.g.,
0.7 Hz, or 3.1 Hz, fib ± f) in the Fourier spectrum. These features are
familiar characteristics of forced nonlinear oscillators. The
ISI return map of the same neuron is shown in Fig.
2G. Long interspike intervals (e.g., those between successive bursts) now range from 0.15 to 0.5 s, resulting in long
bands parallel with the axes. However, a compact cluster associated
with the intraburst spikes is still visible near the origin of the map.
The periodic forcing had only a minor effect on the intraburst spike
pattern while strongly affecting the burst cycle periods.
The PD neuron was found to be far more flexible in response to the sinusoidal current injection when the AB was killed [n = 110 (3)]. Bursts of action potentials appeared during the depolarizing parts of the current stimulus; thus the firing pattern of the neuron became synchronized with Isin(t) (Fig. 3E), corresponding to 1:1 phase locking of activity. This entrainment occurred over a wide range of forcing frequencies (0.2-4.0 Hz). Fourier spectra calculated from the SDFs consist of a large peak at the position of the forcing frequency with a few higher harmonics (Fig. 3H). Interestingly, the minor peak at the position of the gastric frequency (0.2 Hz) is still observable, demonstrating that neither killing the AB neuron nor the sinusoidal forcing eliminated or altered the gastric mill modulation. The return map constructed from the ISIs of this forced PD neuron is now three-clustered (Fig. 3G), similar to that seen with spontaneously bursting PDs in the intact pacemaker group. The cluster associated with the intraburst spikes, however, is more spread (0.04-0.15 s), showing that spike timing is less precise in this situation.
The firing pattern of the forced isolated AB neuron resembles that of the forced PD in the intact pacemaker group. Complex burst patterns emerged due to the interaction between the forcing and the intrinsic oscillation [Fig. 4E; n = 99 (2)]. The periodic forcing affected the timing of bursts more than that of the individual spikes within the bursts. The ISI return map is clearly three-clustered with a compact cluster at 0.01 s (Fig. 4G). Interburst intervals are spread between 0.2 and 0.4 s. The Fourier-amplitude spectrum of the forced AB neuron is quite rich with several peaks appearing at the position of the intrinsic pyloric rhythm, the forcing frequency, and their linear combinations (Fig. 4H).
Spike timing of forced pacemaker neurons: effect of varying stimulus frequency
We surveyed the firing patterns of the pacemaker neurons by
changing the frequency and amplitude of the forcing current in small
steps of 0.1 Hz and 0.5 nA, respectively. As part of the analysis, we
calculated and compared ISI sequences from spontaneously active and from forced neurons. The effect of
Isin(t) on the burst/spike timing of a PD neuron in the intact pacemaker group is shown in Fig.
5A. In control periods,
I(t) = 0, the long interspike intervals show
minor variations as a result of regular pyloric rhythm and stable burst
cycle periods. On the contrary, complex ISI patterns arise
when the neuron is responding to
Isin(t) (gray sections of
Fig. 5A). The resulting ISI sequences have
multimodal distributions. For larger f, 1:1 phase locking
occurs when the neuron receives currents with f fib, at 1.0 and 1.2 Hz here. As the
precision of the burst timing increases, variations in the interburst
intervals become smaller. The timing of intraburst spikes is far less
affected by Isin(t) than
that of the bursts. At f higher than
fib, a characteristic "skipping"
behavior takes place. Many of the bursts of the PD are synchronized
with the forcing current, being generated in the depolarized part of
the sinusoidal waveform. Then, intermittently, a single burst is
skipped, resulting in a lengthened interburst period (Fig.
5A, f = 1.6 Hz).
|
In contrast to the intact pacemaker group, when AB was killed, the PD neurons responded to periodic forcing with less precise and reproducible burst timing (Fig. 5B). In control periods, the ISI values are scattered, in contrast with the bimodal distribution characteristic of normal bursting neurons (Fig. 5B). Definition of bursts is troublesome in such neurons by using interspike data only. Isin(t) injection altered the firing pattern of the PD neuron and induced nearly periodic bursting resembling normal pyloric activity. However, the burst timing of forced PD neurons was far less precise in preparations without AB than in the intact pacemaker group. This is clearly shown in Fig. 5B, where long interspike intervals are randomly distributed in a wide (0.2 s) region.
The ISI pattern of the isolated AB neuron was frequency dependent and multimodal, similar to that of the PD in intact pacemaker group, but more "noisy." The ISI values of Fig. 5C are rather scattered; however, the different modes of synchronization and the skipping behavior (at 3.6 Hz) is still distinguishable.
Dynamics of burst timing and Fourier analysis of spike density data
Frequency-dependent effects on burst patterns were studied using
BCP return maps as well as Fourier transforms of the spike density functions. A wide variety of responses occurred in pacemaker neurons forced at different frequencies. A typical example of the
effects is shown in Fig. 6, where a PD
neuron in the intact pacemaker group was stimulated using
Isin(t) at seven different frequencies. The patterns in the [BCP(i),
BCP(i + 1)] plane display a gradual
transformation as f is increased. At f fib, loops and fixed points are observed
(0.3, 0.4 Hz). The loops are indicators of quasiperiodic behavior. At
f = 1/2fib, 1:2
phase locking occurs, i.e., two bursts are generated during one period
of the forcing waveform. Here a short burst cycle period is followed by
a longer one, resulting in two compact densities on the return map,
symmetrical to the 45° axis. Asymmetric return maps appear at
moderate forcing frequencies (0.7, 0.8 Hz), slightly below that of the
intrinsic bursting. Here a complex, possibly chaotic behavior emerges.
However, the shapes appearing on the return map are still well defined and compact. As a result of the slight irregularities and intrinsic noise present in the pyloric neurons, various phase-locking regimes can
develop, and the burst pattern of the PD often jumps from one to
another, e.g., from a 3:4 to a 4:5 mode and back. One-to-one phase
locking is characterized by a very compact single fixed point on the
diagonal of the graph when we set f = 1.2 Hz, close to
that of the intrinsic oscillation. Skipping or intermittent behavior
results in a three-clustered return map at higher stimulus frequencies
(1.6 Hz), and then 2:1 phase-locking bursting can be observed (not
shown). Development of irregular, possibly chaotic responses was also
observed at yet higher stimulus frequencies (f
3 Hz). A remarkable feature of all our BCP return maps is that each displays well-defined and compact forms (attractors), and has
"forbidden areas" with no points inside.
|
Fourier spectra from the corresponding SDF data are displayed in Fig. 6B. At f below fib, the spectra are dominated by a number of distinct harmonics. In all cases we find a major peak close to the position of fib and another one at f. The positions of the smaller peaks appear as linear combinations of these two fundamental frequencies. Peaks evenly distributed along the frequency axis indicate precise n:m phase locking, e.g., at 0.4, 0.5, or 1.2 Hz. "Noisy," broadband Fourier-spectra (indicators of aperiodic or chaotic response) are obtained in the region below the 1:1 phase-locked mode (e.g., at 0.8 Hz here).
Without AB the response of PD to Isin(t) was far less rich and complex than that of the intact pacemaker group. Isin(t) induced periodic changes in the firing rate of the neurons rather than altering the timing of well-defined bursts (Fig. 7). At low f the arrival times of the irregular bursts show large, irregular variations resulting in dispersed point clouds on the BCP return maps (Fig. 7A, 0.5-0.7 Hz). A single fixed point appears for f = 0.8 Hz. The Fourier spectra calculated from the corresponding SDFs show a similar tendency: a noisy baseline is present at lower frequencies (Fig. 7B, 0.5-0.7 Hz), then clear synchronization is indicated by a large peak (asterisk). One-to-one phase locking is clearly seen in each graph, and the dominant peak always appears at f. Minor peaks appear as higher harmonics. As noted above, a small peak appears at 0.2 Hz in each panel (diamond). This latter component is a result of the intrinsic gastric or esophageal modulation.
|
The autonomous firing pattern of the isolated PD neurons developed from nearly tonic spiking immediately after photoinactivation of the AB to more bursty activity. Nevertheless, the observed synchronization effect was virtually independent of the spontaneous firing pattern of the isolated PD. The 1:1 phase locking was achieved equally when the PD neuron exhibited nearly tonic spiking or more regular bursting. The BCP graphs and Fourier analysis revealed flexible responses of isolated PDs in a wide range of frequencies and amplitudes of the applied current.
The response of the isolated AB neuron to changing f was in several aspects similar to that of the intact pacemaker group (Fig. 8). The BCP return maps show a gradual transformation as the frequency of the injected current increases. One can identify quasiperiodic responses (0.5, 0.9 Hz), a near 1:2 phase-locked pattern, the complex dynamics at 2.0 Hz and 1:1 phase-locked behavior at 3.5 and 4.0 Hz. Here the intrinsic rhythm was unusually fast, close to 3 Hz (Fig. 8B, control). Fourier spectra show complex multiplets of peaks appearing at positions of the pyloric rhythm, the forcing frequency, and their linear combinations. The low-frequency gastric/esophageal modulation is also detectable (diamond).
|
Effects of changing the amplitude of the forcing current
To examine how the synchronization effect depended on the intensity of the periodic forcing, we changed the amplitude A of the injected current while holding the frequency f at constant values. We varied the amplitude of Isin(t) between 0.5 and 2.5 nA. The firing patterns exhibited a strong dependence on the amplitude of the forcing both in PD neurons in the intact pacemaker group and in isolated AB. Figure 9A shows the effect of different A and f on the amplitude ratio of a forced PD neuron (AB intact). At A = 0.5 nA, only a slight modulatory effect is observed. In this case the bursting of the pyloric cells remains quite regular; the position of the peak of intrinsic bursting is unchanged and no phase-locking behavior is observed throughout the range of applied f. Consequently, the Fourier amplitude ratio of the forcing peak at all scanned frequencies is far less than that of the intrinsic oscillation. The Fourier ratio of the forcing is virtually independent of the applied frequency. A slight increase in the amplitude ratio is observed when setting the forcing frequency close to that of the pyloric bursting (f = 1.6 Hz, fib = 1.7 Hz here, arrow). In general, the Fourier spectra calculated from the SDFs of PD neurons receiving low-amplitude current forcing resembled those obtained from spontaneously active pacemaker neurons receiving the intrinsic gastric/esophageal modulation (at 0.2-0.3 Hz). Next, as A is increased to 1.0 nA, the amplitude ratio of the forcing becomes larger, and 1:1 phase-locking appears at f = 1.6 Hz. Phase locking is indicated by the prominently high value seen in the panels (Fig. 9A, 1.0-2.0 nA). Simultaneously, the amplitude ratio of the intrinsic peak vanishes. What we observe here is the merging of the peak of the intrinsic rhythm with that of the forcing, and the two distinct peaks are replaced by a single large peak. Moreover, as the amplitude of the injected current is increased, the amplitude ratio of the forcing becomes larger at all applied frequencies, while that of the intrinsic peak becomes smaller.
|
The behavior of the isolated AB was similar to that of PD with intact AB (Fig. 9B). The amplitude ratio of the stimulus frequency increased with the intensity of the forcing, while that of the intrinsic frequency changed in an opposite manner. Phase-locking (1:1) occurred at f = 2.4 Hz with A = 0.5 nA first, then at 3.0 Hz, when A = 1.5 nA. The amplitude ratio of the forcing was a monotonously increasing function of the current amplitude at all frequencies.
This kind of analysis was performed using the data from PDs of the intact pacemaker group and isolated AB neurons. Since the isolated PD neurons showed no autonomous bursting after killing the AB, no pyloric peak appeared in the Fourier spectra. Phase-locking behavior (1:1) developed in a wide range of frequencies (0.2-3.0 Hz). This phenomenon was virtually independent of the amplitude of the injected current. The isolated PD neurons developed an irregular bursting behavior some time after killing the AB (>1 h). The Fourier spectra of the SDFs of such neurons contained a wide and "noisy" peak and broadband baseline. Nevertheless, the injection of sinusoidal current removed the wide intrinsic peak and resulted in spectra similar to those in Fig. 7.
Various modes of activity of the PD/AB neurons are displayed on
the frequency-current amplitude map of Fig.
10. Here, the frequency of the
sinusoidal current was normalized to the frequency of the intrinsic
(pyloric) bursting. Each symbol represents a single experimental trial
and corresponds to the different modes of operation. This presentation
of the responses is called a phase-diagram or Arnol'd-map
(Glass et al. 1986; Hayashi and Ishizuka
1995
). The zones of 1:1 and 1:2 phase locking have a conelike
shape, the Arnol'd tongue. Each phase-locking zone possesses its own
Arnol'd tongue, but here, for practical reasons, only two are
displayed. Complex dynamics and aperiodic patterns occur between the
tongues.
|
The Arnol'd map of the isolated AB neuron is shown in Fig.
10B. The overall response of the isolated AB neuron is
similar to that of the whole pacemaker group, but the zones of
synchronization are slightly wider here. Although the indicated
frequencies are normalized values, i.e., frequency of the intrinsic
bursting of each neuron was taken into account, the current values are
not normalized. Since the same amount of current can evoke slightly different responses in neurons of the same type (e.g., PD) but from
different animals, it is difficult to define the exact shape of the
tongues in the averaged graph. However, these maps show similarities to
those obtained from "noiseless" theoretical models (Aihara
et al. 1984; Stiber et al. 1997
).
Responses of the electronic model neuron to periodic forcing
The electronic model neuron was subjected to the same type of periodic forcing, as the biological neurons. The autonomous periodic bursting pattern of the four-dimensional EN is shown in Fig. 11A. The frequency of the pyloric-like oscillation was set close to 2 Hz (Fig. 11D). Due to the precisely periodic character of the pattern, the ISI return map consists of only six points, equal to the number of spikes emitted per burst (Fig. 11C). Actually, the six "points" are clusters of overlapping points. When driving the EN with Isin(t) with A = 2 nA, f = 1.4 Hz, the burst pattern became slightly irregular and both the timing of the bursts and the number of emitted spikes were influenced. This is clearly seen as variations of the SDF in Fig. 11F. The ISI return map of the forced EN is similar to that of the forced PDs or AB, i.e., the variations in the long interburst values increased, but the short ISI cluster remained more or less unaltered (Fig. 11G). The Fourier amplitude spectrum of the forced EN has a broadband baseline and several peaks as a consequence of the interplay between the intrinsic rhythm and the forcing (Fig. 11H).
|
The frequency dependence of the forcing effect is shown in the Fig. 12. BCP return maps (Fig. 12A) reveal subharmonic and superharmonic synchronization (0.5, 1.0, and 2.0 Hz) or chaos (0.4 and 1.4 Hz), depending on the ratio between the f and the intrinsic frequency. Accordingly, Fourier spectra contain either equidistant peaks or linear combinations of the two fundamental frequencies with broadband baseline. The response at 2.6 Hz resembles the skipping behavior of the forced PD neuron in the intact pacemaker group to the extent that there are three densities of points; however, these are arcs rather than well-defined and separated clusters. Clear quasiperiodic or skipping (escape) responses were not observed in the EN; instead, high-order n:m synchronization with very long periods or chaotic responses were found. We explain this observation as a consequence of the low amount of intrinsic noise and high stationarity of the EN. Switching between different n:m modes and skipping behavior were frequently seen in lobster pacemaker neurons, while the dynamics of the EN often evolved along a chaotic attractor.
|
![]() |
DISCUSSION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
There is increasing evidence that neural networks at
different levels of complexity can operate in a coherent fashion as a result of synchronization (Destexhe et al. 1996;
Golomb et al. 1994
; Gray et al. 1989
;
MacLeod et al. 1998
). Even in individually irregular
neurons, synchrony and entrainment have been observed (Elson et
al. 1998
, 1999
; Rabinovich et al.
1997
). Experimental studies as well as computational models
show that single neurons can exhibit a wide variety of behavior
including synchronization of bursts/spikes, harmonic or subharmonic
synchronization, quasiperiodicity, and chaotic oscillations
(Aihara et al. 1984
; Hayashi et al.
1982
). Entrainment and chaotic responses have also been
demonstrated in very complex systems such as large populations of
neurons with complex interconnections (Hayashi and Ishizuka
1995
).
In this paper we have experimentally investigated synchronization phenomena in a small network of electrically coupled neurons. Detailed understanding of synchrony and regularity in oscillations of this kind of network is important and required to interpret mechanisms of information processing in such networks as well as in larger ones. Since CPGs play a substantial role in homeostatic regulation of an organism, it is important to explore and interpret the responses of such networks to external perturbations or endogenous rhythmic drives.
The pyloric pacemaker group of the stomatogastric nervous system offers
an especially propitious opportunity for studying synchronization
phenomena, since the architecture and function of this circuit is well
described. This means that detailed modeling of the circuit is possible
both in numerical simulations and in analog electrical circuitry
(Pinto et al. 2000; Sz
). Furthermore, as these networks and their ability to
produce rhythmic output from a collection of component neurons with
complex dynamics becomes understood, similar approaches can be carried out in realistic interpretations of larger networks in vertebrates.
Intrinsic oscillations and periodic inputs in neuronal networks
There have been earlier studies on the stomatogastric nervous
system involving periodic stimulation of nerve cells. Ayers and
Selverston (1979) used existing synaptic inputs of both
excitatory and inhibitory type to stimulate the pacemaker neurons in a
periodic manner (the pacemaker group was not pharmacologically
isolated). The synaptic stimulation elicited large variations in the
arrival times of the pyloric bursts, and a relatively narrow 1:1
synchronization zone was found. Other studies involved intracellular
current injection of rectangular waveforms. Hooper
(1998)
revealed a clear graded relationship between the burst
delay of pyloric PY neurons (in 1:1 synchronized mode) and the temporal
parameters of the waveform defining the rectangular waveform. These
results suggest that the PY neurons can transduce temporal patterns
into neural code. Elson and coauthors (Elson et al.
1999
) used synaptically isolated lateral pyloric
neurons and examined the dependence of the regularization effect on the
polarity and temporal characteristics of the injected rectangular
waveform. To our knowledge, various types of entrainment of neuronal
bursting activity by sinusoidal current injection have not previously
been demonstrated either in the crustacean pyloric pacemaker network or
in any other similar small group of electrically coupled neurons.
In a few experiments we compared the effect of sinusoidal current
waveforms to that of rectangular pulse trains used in earlier studies.
We took the frequency and amplitude of the rectangular pulses to be
equivalent to those of the sinusoids. We found that the extent and
precision of the effect on burst timing and spike density was
significantly greater with sinusoidal waveforms than with rectangular
pulse trains. Fourier spectra of pyloric neurons under stimulation with
rectangular waveforms were more broadband and the harmonics less well
defined. It is known that synaptic neurotransmission between
stomatogastric neurons is of a graded type (Graubard et al.
1980), thus the synaptic currents received by the nerve cells
contain slowly varying components (Manor et al. 1997
).
The main feature of the sinusoidal current waveform may well be its
lack of the high-frequency components found in the rectangular pulses.
In about half of our preparations we detected a characteristic
low-frequency component in the firing patterns of pyloric pacemaker neurons. This modulation possibly originates from an inhibitory coupling from the gastric MG neuron to the AB/PD assembly
(Clemens et al. 1998). Since PTX as a blocker of fast
inhibitory connections was used in all of our experiments, and the
gastric modulation was present, this inhibitory synapse is probably not
of the fast glutamatergic type (Cleland and Selverston
1998
).
Our results also raise an issue regarding the interaction between
distinct periodic oscillatory pattern generators. As seen in many
experiments, the low-frequency and low-amplitude sinusoidal current did
not worsen the precision of the intrinsic pyloric oscillation; rather,
a subharmonic (e.g., 1:5) synchronization took place. Analogously, the
low-frequency gastric modulation can act as a regularizing factor on
the pyloric rhythm. The reverse effect, i.e., the regularization of the
gastric rhythm by the fast pyloric oscillation has already been
elegantly demonstrated in crabs (Bartos et al. 1999).
Variations in the timing of pyloric bursts as well as in the
alterations in the number of emitted spikes are possible ways of
information processing in the pyloric CPG. The various temporal patterns of burst activity revealed by return maps show stable frequency-dependent output responses to periodic forcing. These temporal forms displayed notable long-term stability and consistency during the periodic forcing, especially in the intact pacemaker group.
In this respect, the temporal forms and various entrainment zones were
similar to those found in periodically driven PY neurons described by
Hooper (1998). The pyloric pacemaker neurons transduce incoming temporal patterns (here, sinusoidal waveforms) into stationary neural codes, thus representing a form of information storage (Golowash et al. 1999
; Matsugu et al.
1998
).
In our experiments, intermittent or chaotic responses were observed only in a narrow range of stimulus parameters showing that the intact pyloric pacemaker can maintain a long-term stable oscillation and has a tendency to express periodic patterns even in the presence of such external perturbations. Termination of the current injection quickly led to restoration of the original pyloric rhythm in the intact circuit, i.e., the burst patterns were virtually indistinguishable before and after the forcing. Even extended (up to 15 min) periodic forcing did not induce any aftereffects in the firing patterns, phase shifts, or reconfiguration of the intact pyloric network. It is indeed one of the important observations of the current work that the rhythmicity and stationarity of the pyloric oscillation is well preserved under the influence of long-term external periodic perturbations. In these circumstances the pyloric network shows no signs of plasticity induced by the forcing.
Intact pacemaker group versus reduced configurations
The spontaneous behavior of the pyloric pacemaker group as well as
its response to periodic forcing were completely different when the
main pacemaker neuron AB was present and when it was removed by
photoinactivation. It appears then that rich dynamical behavior,
phase-locking, numerous zones of synchronization, and quasiperiodicity
are characteristics of the intact pyloric pacemaker group only. When AB
is killed, the PD neurons remain able to spike and burst, but in a more
irregular manner than in the intact pacemaker group (Bal et al.
1988; Elson et al. 1998
). Their activity can evolve from tonic firing, through irregular spike generation, to clear
(but still irregular) bursting. The basis for this change in activity
is unclear. It may reflect recovery from nonspecific damage (with
increased leakage currents) following the death of the coupled AB cell;
alternatively, it could also result from an intrinsic,
activity-dependent modulation of membrane conductances (Turrigiano et al. 1995
). The time course of the
transformation of firing patterns in isolated PD neurons (hours) is
faster than that in cultured stomatogastric neurons (days). Regardless
of their activity pattern, however, the remaining PD neurons show simplified, flexible responses (phase locking) to external stimuli. On
the other hand, when the PDs are killed, AB retains strong oscillatory
properties (although with less stability and stationary than the intact
pacemaker). Compared with the isolated PDs, AB shows a narrower range
of phase locking. The combination of rich dynamical function and
flexibility therefore depends on the cooperative behavior of the three
neurons. AB is the robust part of the pacemaker, while the two PD
neurons are clearly the flexible ones.
Taken together, the intact pyloric pacemaker group of neurons is functioning as an optimized, single, low-dimensional oscillator capable of both initiating a stable motor pattern and responding to time-varying signals. The intact group exceeds in performance all other reduced configurations of the pacemaker neurons.
Motivations and benefits of the spike density function technique
In our study we analyzed in detail spike time data, while raw
membrane potential time series were only visually inspected. This
approach is supported by several arguments. First, spikes, rather than
slow-wave membrane oscillations control the output of pyloric muscles.
The evolution of the membrane potential of single neurons is of small
significance in controlling muscle activity. The motor response depends
on the number and temporal patterns of spikes of the presynaptic
neurons (Morris and Hooper 1997). Second, there is a
roughly monotonous relationship between the instantaneous firing rate
of the neuron and the actual membrane potential meaning that the active
depolarized periods of bursts can be clearly identified by knowing the
arrival times of spikes or equivalently, the instantaneous firing rate.
Spike density therefore suitably represents the bursting activity of
pyloric cells or other types of neurons displaying plateau potentials.
The computed SDF function reveals fine modulations in the firing pattern, which are not easily recognizable in the original time series (e.g., gastro-pyloric interaction). Fourier analysis of the SDFs provides a further efficient way to characterize the spike trains and interactions between rhythmic networks. The amplitude and width of the peaks as well as the overall shape of the spectrum clearly shows the most important temporal features of the firing patterns.
Variations in the input resistance and size of the pacemaker neurons as well as the relative position of the current injecting electrode made it somehow difficult to average and statistically analyze data obtained from different animals. Zones of synchronization and irregular dynamical behavior were evident and well defined when altering the frequency and amplitude of Isin(t) using the same preparation and configuration. However, the magnitude of the effects varied among different preparations. Normalization of the current frequency was straightforward across preparations as the frequency of the intrinsic pyloric oscillation offered a reference value for each measurement. Similar simple normalization technique for the current amplitude (or current density) could not be performed.
Low dimensional oscillatory dynamics in models and biological neurons
Despite the enormous biophysical and biochemical complexity of
living neurons, they can express low-dimensional behavior. In this case
the evolution of the membrane potential, the most commonly measured
state variable of the neuron, depends on the interaction of a small
number (3-5) dynamical variables (Falcke et al. 2000;
Rabinovich et al. 1997
). Equivalently, the time
evolution of the membrane potential and the other variables can be
described using the same number of differential equations; however, the exact form of the equations might be unknown. The behavior and interactions of individual neurons are still determined by a large number of cooperating biophysical and metabolic processes. These processes, however, can be combined into a small number of abstract dynamical variables, like those appearing in the Hindmarsh-Rose model.
Bursting activity of motor pattern generating neurons can well be
described using the simplistic polynomial models. Although the
cooperative behavior of the network is a periodic oscillation, some of
the individual neurons, when isolated from their counterparts, can
display chaotic firing patterns (Elson et al. 1998
;
Rabinovich et al. 1997
).
Our earlier results have shown that a simple four-dimensional model
reproduced significant functional aspects of individual pyloric neurons
with regard to membrane voltage activity. Depending on the setting of
internal coefficients and current offsets, the EN can generate periodic
or chaotic bursting or tonic spiking. In further experiments this EN
displayed a remarkable ability to interact cooperatively with living
pyloric CPG neurons: the electronic model neuron was connected to
synaptically isolated PD neurons via dynamic clamp, and together they
produced regular oscillatory patterns similar to those in intact
pyloric networks (Sz). The
experiments reported here give further support to the use of
low-dimensional polynomial models to simulate networks of biological neurons.
![]() |
ACKNOWLEDGMENTS |
---|
We thank R. Pinto and G. Stiesberg for constructing and calibrating the electronic neuron model.
Partial support for this work came from the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Engineering and Geosciences, under Grants DE-FG03-90ER14138 and DE-FG03-96ER14592, and from the Office of Naval Research under Grant ONR N00014-00-1-0181.
![]() |
FOOTNOTES |
---|
Address for reprint requests: A. Sz
Received 24 May 2000; accepted in final form 18 December 2000.
![]() |
REFERENCES |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|