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 Previous Article
The Journal of Neuroscience, April 15, 2000, 20(8):3041-3056
Modeling of Spontaneous Activity in Developing Spinal Cord Using
Activity-Dependent Depression in an Excitatory Network
Joël
Tabak1,
Walter
Senn2,
Michael J.
O'Donovan1, and
John
Rinzel3
1 Laboratory of Neural Control, National Institute of
Neurological Diseases and Stroke/National Institutes of Health,
Bethesda, Maryland 20892, 2 Physiologisches Institut,
Universität Bern, CH-3012 Bern, Switzerland, and
3 Center for Neural Science and Courant Institute of
Mathematical Sciences, New York University, New York, New York 10003
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ABSTRACT |
Spontaneous episodic activity is a general feature of developing
neural networks. In the chick spinal cord, the activity comprises episodes of rhythmic discharge (duration 5-90 sec; cycle rate 0.1-2
Hz) that recur every 2-30 min. The activity does not depend on
specialized connectivity or intrinsic bursting neurons and is generated
by a network of functionally excitatory connections. Here, we develop
an idealized, qualitative model of a homogeneous, excitatory recurrent
network that could account for the multiple time-scale spontaneous
activity in the embryonic chick spinal cord. We show that cycling can
arise from the interplay between excitatory connectivity and fast
synaptic depression. The slow episodic behavior is attributable to a
slow activity-dependent network depression that is modeled either as a
modulation of cellular excitability or as synaptic depression. Although
the two descriptions share many features, the model with a slow
synaptic depression accounts better for the experimental observations
during blockade of excitatory synapses.
Key words:
spontaneous activity; oscillations; depression; developing spinal network; recurrent excitation; rate model
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INTRODUCTION |
Spontaneous activity generated by
networks of synaptic connections is a characteristic, but poorly
understood, feature of the developing nervous system. It has been
detected in many different regions, including the retina, the
hippocampus, and the spinal cord. In each region, bouts of activity
occur episodically, separated by periods of quiescence. Although the
mechanisms responsible for this activity are incompletely understood,
it is known that developing networks are hyperexcitable, in part,
because the classically inhibitory neurotransmitters, GABA and glycine,
are depolarizing (Cherubini et al., 1991 ; Sernagor et al., 1995).
We have been studying this activity in the isolated lumbosacral spinal
cord of the chick embryo. This preparation generates spontaneous
episodes of activity in which spinal neurons are cyclically activated
(see Fig. 1). This activity is not produced at a specific location, nor
by a specific set of connexions (Ho and O'Donovan, 1993), and is still
present in networks for which glutamatergic transmission or
glycinergic/GABAergic transmission has been blocked (Chub and
O'Donovan, 1998a). It has also been shown that episodes of activity
transiently depress network excitability, which recovers in the
quiescent phase; this depression is manifest as decreased synaptic
strength (Fedirchuk et al., 1999) and hyperpolarization of individual
cells. These observations have led to the proposal that the occurrence
of episodes is a population phenomenon that is controlled by
activity-dependent network depression (O'Donovan and Chub, 1997;
O'Donovan, 1999). There is evidence that spontaneous bursts or
episodes in other parts of the nervous system are also regulated by a
form of activity-dependent depression. A slow "refractoriness" is
thought to restrict wave generation and propagation in the newborn
ferret retina (Feller et al., 1996, 1997; Butts et al., 1999). In
hippocampal networks, NMDA receptor desensitization (Traub et al.,
1994) or presynaptic depletion of glutamate pools (Staley et al., 1998)
have been proposed to regulate the occurrence and/or duration of
synchronous bursts.
Less is understood about the mechanism of cycling within an episode,
but it has been proposed that this could also be regulated by
activity-dependent depression with much faster kinetics than the
depression controlling episodes [O'Donovan and Chub (1997); see also
Streit (1993)]. Recently, it has been shown in modeling studies that a
purely excitatory network with random connections susceptible to fast
depression is capable of generating oscillations similar to the cycling
observed during episodes (Senn et al., 1996; J. Streit and W. Senn,
unpublished results).
Using these ideas, we develop a simple and general model of network
activity. This model describes only the average firing rate of the
population, rather than the membrane potential and spikes of the
individual neurons. It assumes a purely excitatory recurrent network
that is susceptible to both short- and long-term activity-dependent
depression (see Fig. 2A). The model is described by only
three coupled nonlinear differential equations. Therefore, the analysis
can be done graphically, step by step, which facilitates an intuitive
understanding of the network dynamics.
We show that this simple model can account for the occurrence of
spontaneous episodes, the rhythmic cycling within an episode, and the
developmental changes in the duration of episodes and interepisode
intervals. Most importantly, it shows that the recovery of spontaneous
activity in the presence of excitatory amino acid blockers (Barry and
O'Donovan 1987; Chub and O'Donovan 1998a) is a property of
networks with a slow form of synaptic depression.
Parts of this work have been presented previously in abstract form
(Rinzel and O'Donovan, 1997) and conference proceedings (O'Donovan et
al., 1998; Tabak et al., 1999).
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MATERIALS AND METHODS |
Experimental
All experiments were performed on the isolated spinal cord of
White Leghorn chicken embryos maintained in a forced-draught incubator. The spinal cord was dissected out under cooled (12-15°C) oxygenated (95% O2, 5% CO2)
Tyrode's solution containing (in mM): 139 NaCl, 12 glucose, 17 NaHCO3, 2.9 KCl, 1 MgCl2, 3 CaCl2. The preparation was
transferred to a recording chamber (at room temperature) and superfused
with oxygenated Tyrode's solution that was later heated to 26-28°C.
The concentration of KCl was raised to 5-8 mM during some
of the recordings to accelerate the recovery of activity after
glutamatergic blockade (Chub and O'Donovan, 1998a).
Neural activity was recorded from ventral roots or muscle nerves.
Recordings were made using tight-fitting suction electrodes and
amplified (DC 3 kHz) with high gain DC amplifiers (DAM 50 and IsoDAM,
World Precision Instruments). Amplified signals were directly digitized
through a PCI board (National Instruments) and/or recorded on tape
(Neurodata) for further analysis. Recordings were digitized at a low
sampling rate (5-20 Hz) because the events of interest occur on a slow
time scale (Fig. 1).
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Figure 1.
Spontaneous episodes of activity recorded from
ventral roots of chick embryo spinal cord in vitro at
embryonic day (E) 7.5 and 10. Recordings were made in DC mode and
digitized at 10 Hz, showing the synaptic depolarization recorded
electrotonically from motoneurons. Periods of activity (episodes) are
clearly distinct from silent phases. An interval is defined as the time
between the beginning of two consecutives episodes. Episodes are
composed of cycles, or "network spikes." With development, both the
interval and episode duration lengthen, and cycle frequency and the
level of depolarization (tonic component) also increase. Also, on older
embryos episodes have a leading tonic phase before cycling begins. For
the examples shown here, interval and episode durations are,
respectively, E 7.5: 440 ± 29 and 20 ± 1 sec; E 10:
1600 ± 140 and 68 ± 3.5 sec (mean ± SD).
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Theoretical
Model equations. The two models are described and
analyzed in depth in Results. Here we present them and the associated
parameter values more succinctly. Both models have three dynamical
variables, including a and d: a is the population
activity or mean firing rate, and d is the fast depression
variable, representing the fraction of synapses not affected by the
fast synaptic depression. These two variables generate the cycles
within an episode. The slow modulation that underlies onset and
termination of episodes, and the long silent phases, is described by
 or s. represents the threshold for cell firing. In
the -model, it increases slowly during an episode, eventually
terminating the episode and recovering between episodes. In the
s-model, s represents the fraction of synapses
not affected by the slow synaptic depression, decreasing slowly during
an episode, terminating it, and then recovering between episodes. In
the s-model, the fraction of nondepressed synapses at any
given time is therefore the product s·d.
The equations for the models are:
-model:
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(1)
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(2)
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(3)
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s-model:
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(4)
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(5)
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(6)
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The parameter n measures the connectivity in the
network (see Results). All of the x
functions are sigmoidal (e.g., see Fig. 2C for a plot of
a):
The parameter x defines the argument's value for
which half-activation occurs:
x(x) = 1/2. The value of
kx (positive) determines the sigmoid's
steepness, smaller values being steeper. Note that the presence
(respectively absence) of a minus sign in front of (a  x) indicates that the sigmoid is increasing (respectively decreasing). The variables d and
s are high for low activity (low depression) and low for
high activity (high depression), so d(a) and
s(a) are decreasing functions of
a.
Although a is the cellular input-output
function, it should be considered as an average over the population of
neurons. Assuming these neurons are not identical,
a inherits a sigmoid shape; can be
interpreted as the mean threshold and ka as the
threshold variance, assuming a unimodal distribution of thresholds
(Wilson and Cowan, 1972). Because the connectivity coefficients are
included inside the a function, a
should be interpreted as mean population firing rate averaged over a brief period of time, not as instantaneous voltage (Pinto et al., 1996); a(n·s·d·a) is the instantaneous
firing rate. Note that because a(0) is not
equal to zero, we are implicitly assuming that there is some background
activity or fraction of spontaneously active neurons in the network.
However, the model does not depend on this feature. Indeed, if
a(0) = 0, similar episodes of activity
are produced, except that a stays virtually at 0 from just
after an episode to just before the next one (our unpublished results).
Unless stated otherwise, we use the parameter values given in Table
1.
Simulations. The model equations were implemented within
XPPAUT (freely available software by G. B. Ermentrout,
http://www.pitt.edu/~phase/), a general purpose
interactive package for numerically solving and analyzing differential
equations. XPPAUT includes a tool for calculation of bifurcation
diagrams (AUTO). Simulations were performed using the Runge-Kutta
integration method with a time step of 0.2 (dimensionless units). We
checked that the results were unchanged when the time step was halved.
Simulations were run on Unix PCs. A version of the software WinPP runs
under Win95 or Win98.
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RESULTS |
We have argued elsewhere (Ritter et al., 1999) that the spinal
network responsible for spontaneous activity is composed of recurrently
connected, functionally excitatory synaptic connections. Furthermore, because the expression of network activity does not appear
to depend on the details of network architecture or membrane properties
and because detailed biophysical information on cell and coupling
properties is unavailable, we are justified in using a very idealized
model. Throughout, we use mean field models that describe only the
averaged population behavior, not the detailed electrical activity of
individual cells. The models are also temporally coarse-grained so that
action potentials are not represented per se, only firing rate.
Our graphical analysis of the models proceeds by considering reduced
models, beginning with the fastest process first and expanding the
model by stepwise introduction of the next slower variables. We begin
with the single Equation 1 that governs the dynamics of a recurrent
excitatory network, without the negative feedback of depression. This
reduced model illustrates the classical behavior of bistability in a
recurrent population, of either a low or high activity state of steady
firing. Next, we introduce a fast activity-dependent synaptic
depression, which can lead to network oscillations (cycling) in this
two-variable system. We analyze this subsystem with phase plane
methods, finding a different type of bistability. Finally, we add slow
activity-dependent depression in either of two forms and show that
these three-variable models can generate episodes similar to those
observed experimentally in the chick embryo spinal cord and other networks.
A recurrent excitatory network can be bistable
In this section, we consider only the dynamics of the average
activity in a random excitatory network; the depression variables are
frozen. We use a graphical method to illustrate how the steady activity
states of the one-variable system are determined. These steady states
depend on parameters such as the network connectivity, and we show that
for certain values of connectivity the network can have two stable states.
Here, and throughout, we let a be a measure of the network
activity, for instance the population's mean firing rate, relative to
the maximum possible rate. Its time behavior, according to classical
descriptions (Wilson and Cowan, 1972), satisfies  a  = a(Ieff) a, where is the time derivative of
a and Ieff is the effective input to
cells in the network. Accordingly, the activity of the network reaches
the value a(Ieff) with a time
constant a. a is called the
activation function and describes how Ieff is
converted into firing rate (Fig.
2B). For small inputs,
a is near 0; very few neurons are firing. For
large inputs, a saturates at 1, with all
cells in the network being fully active. There is a rather sharp
transition between these two limits (if ka is
small) at the value Ieff = for which
a() = 1/2. corresponds to
the average threshold for cell firing. The mathematical expression of
a is given in Materials and Methods.
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Figure 2.
A, Schematic diagram of recurrent network
with random connections. The output from the network is fed back to the
cells. The amount of feedback depends on network connectivity
(n). Activity (a) can also be modulated at the
synaptic level by a fast depression (d) and by a slow
depression (s). The effective input to cells is therefore
n·s·d·a. The resulting postsynaptic firing rate, at
steady state, is given by a(a·d·s). This
postsynaptic firing rate can also be modulated by the cells' threshold
(). Note, our mathematical formulation does not distinguish
presynaptic and postsynaptic effects. B, Cellular
input-output relationship, averaged over the population.
Ieff is the effective input to the network (see
Results). The cellular threshold parameter can also be viewed as a
slowly modulated variable. ka measures the width
of the transition range around threshold; a small value of
ka implies a steep activation function.
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Because of the recurrent connectivity, the input
Ieff is produced by the network's own activity,
in addition to any external inputs. Therefore, in the absence of
external inputs Ieff = n·a, where we introduce the parameter n as a measure of network
connectivity (in arbitrary units). The greater the connectivity, the
greater the recurrent input being fed back to the network for a given amount of activity (n is a gain parameter, measuring how the
average cellular output is transmitted to other cells, and thus depends on many physiological parameters such as the average number of contacts
from a neuron, average synaptic efficacy, etc.). Thus, the dynamics of
this network is described by:
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(1′)
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What is the behavior of the system defined by Equation 1'? After
any transient input, a will evolve to a steady state, for which = 0. Its value at steady state must then
satisfy a = a(n·a). The solutions to
this equation are represented graphically in Figure
3A as the intersections of the
curves f(a) = a and g(a) = a(n·a) for different values of n.
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Figure 3.
Analysis of the single-variable activity model for
a recurrent excitatory network. A, The steady states satisfy
a(n·a) a = 0. They are
determined by the intersection(s) between the curves f(a) = a (thick line) and g(a) = a(n·a)
(smooth curves), shown here for different values of
n. At these intersections a(n·a) a = 0. B, Time course of the activity for n = 0.5. At t = 20 a brief stimulus is applied but
fails to bring the network above threshold (i). Stronger
stimulation applied at t = 50 excites the network above
threshold, so the high activity steady state is reached
(ii). The network will stay at this high level unless a
large perturbation brings it back below threshold. C, Plot
of the steady-state values of a for all possible values of
n between 0 and 1. The dashed portion of the
curve indicates unstable steady states. In A and
C, the steady states for n = 0.5 are
represented by filled (stable) or open (unstable)
circles. The arrows indicate that the activity,
if initially below threshold (unstable steady state), will fall back to
the low stable state (i, < 0), and if it is
initially above network threshold it will increase until it reaches the
high stable state (ii, > 0), as illustrated in
B. = 0.18.
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For small values of n, there is only one steady state, at a
low value of a, with the network barely active because of
its low, functional connectivity. Even if the network receives a strong transient external input, it will return to this quiescent state because the connectivity is too low to sustain the activity. Similarly, for large connectivity, there is only one possible steady state (the
curves do not actually intersect at the low level), for which the
network is very active (a near 1). A more interesting case occurs for intermediate values of the connectivity parameter: there are
three steady states, at low, intermediate, and high values of
a. However, the intermediate steady state is
unstable. That is, a small perturbation from it will grow,
and the network activity will evolve toward one of the two other steady
states. Therefore, the network has two possible stable steady states; it is bistable. If the network is in the low steady state,
it can be switched to the high steady state by a transient stimulus that brings a above the intermediate steady state, which is
the effective network threshold. If the stimulation does not increase the activity above threshold, the network will fall back to low activity after the stimulus. This network threshold is not equal to the
cell threshold ; for example, if a is very
steep, the network threshold approximately equals /n; it
decreases with increasing connectivity. Note that once the network has
switched from the low to high (or high to low) state, it will stay in
the new state even though the perturbation has stopped. It will not return to the previous state, unless a new perturbation of sufficient magnitude occurs (if neural noise were incorporated explicitly in the
model there might be spontaneous transitions between the two states).
This type of bistable behavior, illustrated in Figure 3B
(see figure legend), is characteristic of recurrent excitatory networks
(Wilson and Cowan, 1972). In the next sections, we will find that
bistability underlies our models' episodic rhythms; slow modulation
leads to transitions between the attracting pseudostates of cycling
during an episode and quiescence between episodes.
Figure 3C summarizes the above results by showing the
activity levels at steady state ( = 0) over a
range of n values. Again, it illustrates that for low
connectivity/efficacy (n small) there is only one steady
state at low activity. For strong connectivity (n near 1 and
above) only the high activity state is present. For an intermediate
range of n values both stable states coexist, along with an
unstable state of intermediate activity (dashed line). The
states are continuously connected as you travel along the S-shaped
curve. However, as n is varied over the range, one encounters discontinuous transitions between the stable states, at the
S-curve's knees. This figure, showing the different possible activity
states for all values of a control parameter, is called a bifurcation
diagram [see Strogatz (1994); a bifurcation is a point for which there
is a qualitative change in the dynamics of the system, like at the
knees]. It will be useful to distinguish the three
"branches" of such S-curves: upper, lower, and middle. Here,
upper and lower ones correspond to the stable steady states, and the
middle one corresponds to the unstable steady state. The middle branch
is a separatrix between the upper and lower branches: any state of the
network above it (i.e., on the right side of the S-curve) will evolve
toward the high steady state, and any state below it (i.e., on the left
side of the diagram) will fall to the low steady state. It is easy to
understand why: for any point on the S-curve, by definition,
= 0. If the connectivity is increased (i.e., the
point shifted to the right) even slightly, this increases
a(n·a), and therefore > 0. Conversely, if n is decreased this leads to
< 0.
A recurrent excitatory network with fast
activity-dependent synaptic depression can
produce oscillations
In the above section, we studied the behavior of a recurrent
excitatory network as a function of its (fixed) connectivity. Here, we
will analyze network behavior when the effective connectivity is
changing because of synaptic depression. We will analyze the resulting
two-variable system graphically in the phase plane and show that it can
generate oscillations (like the cycling during an episode) and
bistability, depending on the average neuronal excitability.
As a preliminary step, reconsider Figure 3C and suppose that
we sweep n slowly back and forth over a range that includes
the bistable regime. This would lead to alternating phases of high and
low activity, with sharp transitions between phases. If n were decreasing while the activity is high, the system's state point
would track the upper branch, moving leftward, until we reached the
S-curve's left knee. Then the state point would fall to the lower
branch. The activity would drop precipitously. Now at low activity, if
n were increasing, the system would track the lower branch,
with activity rising very gradually, until the right knee was reached.
Then the state point would jump back to the upper branch of high
activity, completing a cycle, and the process would repeat.
We really want a description so that such cycling happens
spontaneously, rather than being driven as above. With this goal we
make the effective connectivity a dynamic variable by
multiplying n by the variable d, the kinetics of
which depends on activity, guaranteeing that d automatically
decreases when activity is high and increases for low activity. In this
formulation, d is defined as the fraction of nondepressed
synapses from a representative cell to its targets (e.g., the
fraction of releasable transmitter at those synapses). Its value ranges
from 0 (synapses totally depressed) to 1 (all synapses fully
available), so the effective connectivity (n·d) ranges
from 0 to n (n, being an arbitrary measure of connectivity
can be >1). In all of the following figures, we set
n = 1 unless mentioned otherwise. Equation 1' thus
becomes:
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This idealized model does not distinguish between
glutamatergic excitatory synapses or depolarizing
gabaergic/glycinergic synapses.
We use a simple first-order kinetics for d, reminiscent of
the gating kinetics in Hodgkin-Huxley-like models for voltage-gated ionic currents:
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()
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where d(a) is a decreasing sigmoid
function (see Materials and Methods). When the system is at low
activity, d tends to increase, whereas for high activity
d decreases. d tends toward d(a) but lags behind changes in a
with a time constant d. Hence, the system can
potentially oscillate, but whether it does depends on parameter values.
For example, if the negative feedback from depression is too fast, the
oscillations are precluded. The condition (parameter relationship) for
which oscillations emerge in the model is given in the Appendix.
We need to examine the dynamics of this a d system
over a range of values, because in later sections will become a
slow modulatory variable. For small, the network is highly
excitable, and it operates at a steady high-activity level. If the
network is held at a low activity level with synapses fully available and then released (Fig. 4A),
it tends to its desired high level; in this case, it shows damped
oscillations around the steady state because the depression is delayed
(in the same way as the "delayed rectifier" potassium current does
not turn on immediately after a voltage increase).
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Figure 4.
Dynamical behavior of the a d
system for low firing threshold, (0.15). A, Time course
of activity a (solid curve) and depression d
(dashed curve) showing that the system reaches a steady
state after some damped oscillations. B, Phase-plane
representation of the dynamics, showing the nullclines and trajectory
of network state. The a- and d-nullclines are
defined by = 0 and  = 0.
There is only one intersection, therefore the system has only one
steady state. It is a stable steady state; as shown by the time course
(A) or phase-plane trajectory. The damped oscillations
evident in the time course (A) are seen here as spiralling
toward steady state. n = 1.
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The system's time course from Figure 4A can be viewed in
another way, as a trajectory by plotting in the (a, d)
plane simultaneously the points defined by the values of a
and d at consecutive and equally spaced times (Fig.
4B, dotted line). Although time is not represented
explicitly on such a diagram, velocity along the trajectory can be
deduced from the spacing between consecutive points. Here, on release
from low activity the network quickly (large spacing between points)
moves toward a state of high activity with the synapses depressed,
slowing down (small spacing between points) as it approaches the steady
state, in a spiraling fashion (the damped oscillation).
Using the a d plane, we can analyze and predict
qualitatively the system's dynamical behavior and the consequences of
changing various parameters [for an introduction to phase plane
analysis, see Rinzel and Ermentrout (1998) or Strogatz (1994)]. The
power of this graphical/geometrical approach comes largely from
understanding the two solid curves in Figure 4B. These
curves are called the nullclines. The a-nullcline is defined
as the relationship between a and d that is
obtained by setting = 0. Similarly, the
d-nullcline comes from setting = 0.
Given these nullclines, we can map out the flow directions in the phase
plane, according to the different areas delimited by the nullclines.
For example, in the area on the right of the a-nullcline and
above the d-nullcline, we know that a must be
increasing and d decreasing (hence the arrow
pointing up and left). Also, we know that a must be at a
local minimum or maximum when the trajectory crosses the
a-nullcline and so on. Where the nullclines intersect,
neither a nor d is changing, and this corresponds
to a steady state of the system.
The a-nullcline is defined implicitly according to
a = a(n·d·a). For a physiological
interpretation, consider d as a parameter so that points on
the a-nullcline are the steady-state values of a
at which decay of a balances the regenerative input from
recurrent excitation in Equation 1. Of course, the result is similar to
the S-curve in Figure 3C. The d-nullcline is
simply the graph of d(a). The nullclines are
drawn in Figure 4B, showing that there is only one possible
steady state (the unique intersection of the two curves). In general
there can be one or three (exceptionally two) intersections, as we will
show below for some values of . The a-nullcline has the
characteristic switchback form (S-shape) of autocatalysis in nonlinear
excitable and oscillatory systems. Although the steady state is stable
in this case, it could be made unstable by increasing d
to slow the negative feedback process, in which case the activity would
become oscillatory.
For sufficiently large values of , there is again only one steady
state, but now at a low activity level: in Figure
5B, the unique intersection of
the nullclines near the abscissa at d = 0.923
(vertical arrow). Note that the a-nullcline's
low-activity branch is very close to the abscissa. This network model
is excitable. If the system is stimulated briefly but
adequately while at this quiescent, resting state, it exhibits a
transient response of high activity and then returns to rest. Figure
5A shows the threshold effect by plotting a(t)
and d(t) for two different initial conditions. In one case
(gray curves), the activity starts just below threshold and
directly returns to the steady state. In the other case (black curves), the activity is initially just above threshold and then increases in a regenerative way before depression returns it to the
steady state. Note that this threshold for network excitability is
approximately given by the value of the middle branch of the a-nullcline and should not be confused with , which is
the average cellular threshold. This phase plane portrait gives us a
graphical way to see the effect of increased threshold, by comparison
with Figure 4B. The a-nullcline's right shift is
understandable intuitively. For a given value of depression variable
d, one expects with increasing to lose the possibility
of a high-activity state in the a-subsystem. With enough
increase of the high-activity state in the full a d system is precluded.
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Figure 5.
Dynamical behavior of the a d
system for high (0.28). A, Time course for two different
initial conditions. In one case, a(0) is above a threshold,
and the network exhibits a transient high-activity phase and then falls
to quiescence (black curves). In the other case, the network
goes directly to "rest" (gray curves). a is
indicated by the solid curves; dashed curves
represent d. B, Phase-plane representation of the dynamics,
showing trajectories for the two cases illustrated in A.
There is only one steady state, defined by the intersection of the
d-nullcline and the lower branch of the
a-nullcline (for a  0) the quiescent
state. If the system is perturbed away from steady state by a sudden
increase in activity, the threshold for generation of a transient cycle
of high activity is approximately given by the middle branch of the
a-nullcline (it is actually slightly above). n = 1.
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Finally, for some intermediate values of , there can be three steady
states, as shown on Figure 6, A and
B. In both A and B, the lower steady state is stable, and the intermediate
one (very close to the low state) is unstable. However, the stability of the high-level steady state depends on the particular values of the
other parameters. An important parameter, which does not change the
location of the steady states, is d. The high-level steady state is stable in Figure 6A, for a value of
d = 1. Hence, the system is bistable. If the value
of d is increased, the negative feedback caused by
depression is more delayed, which can lead to oscillations around the
steady state (now unstable), as shown in Figure 6B. The
system is bistable again, but now we have a low-activity steady state
coexisting with a high-activity oscillatory state. Depending on the
initial conditions (or external inputs), the system can either reach
the low steady state or the high activity state, either steady (Fig.
6A) or oscillatory (Fig. 6B,C).
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Figure 6.
Dynamical behavior of a d
system for intermediate (0.2). A, Phase-plane
representation of the dynamics for d = 1. The high
and low steady states are stable. Any trajectory eventually reaches one
of them, depending on initial conditions. A perturbation away from the
low steady state can switch the system to the high state, if it is
above the middle branch (black trajectory). Otherwise, the
system falls back to the lower state (gray trajectory).
B, Phase-plane representation of the dynamics for
d = 2. This higher value destabilizes the high
steady state and leads to the appearance of a stable oscillation
(closed orbit) around it. Again, a sufficient perturbation
can switch the system from the low-activity steady state to
oscillations. C, Time courses for the trajectories shown in
B. Superthreshold perturbation leads to an oscillation
(black curves), whereas after a subthreshold perturbation,
activity returns directly to the low steady state (gray
curves). The dashed curves represent d. n = 1.
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We summarize this collection of possible behaviors in a bifurcation
diagram (Fig. 7A) by plotting
the a-values for the steady and oscillatory states when the
parameter is varied. The Z-shaped curve (black solid
line and dashes) for steady states is analogous to the
S-curve of Figure 3C; the orientation is
left-right-reversed because increasing reduces network
excitability, like decreasing n. Starting at low with
the unique high-activity steady state and moving rightward along
this curve, we see that this state loses stability at > 0.181 (solid turns to dashed). At this point, a stable
oscillation emerges, around the steady state. We plot the maximum and
minimum values of a during a cycle (thick gray
curves). At emergence, the oscillation's amplitude is very low,
but then grows as increases. This type of oscillation birth is
called a Hopf bifurcation (Rinzel and Ermentrout, 1998). At a critical
value of (c = 0.207), the oscillation
encounters the unstable state on the Z-curve's middle branch and
disappears. That is, the minimum activity during a cycle is just equal
to the excitation threshold, and the oscillation cannot be maintained if is increased any farther. For larger only the low-activity steady state is stable.
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Figure 7.
A, Bifurcation diagram showing the possible
behaviors of the a d fast subsystem for different
values of . For low values of , there is only one steady state,
at high level, which becomes unstable for increasing . For
intermediate values of (around 0.2), the system is bistable: there
is a stable high-activity oscillation and a stable, low steady state.
With larger values of , the system can only be in a low steady
state. Circles represent the cases shown in Figures
4B, 5B, and 6B. Thick black curves: stable
steady states; dashed, thin curve: unstable
steady states; thick gray curve: maximal and minimal values
of a for the periodic oscillations. B, Period of
the oscillations.
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The system is bistable for between the Z-curve's left knee and
c. Over this range, it can be either at a low, steady
activity level, or at a high, oscillatory level. These high-activity
oscillations mimic the cycling observed during an episode of activity
in the chick spinal cord. In the next sections we will see how slow
autonomous dynamics for this modulatory variable can sweep the
a d subsystem back and forth over this bistable
regime to account for the cord's episodic rhythms. Thus, although
Figure 7A does not show motion per se, it forms the skeleton
over which episodic dynamic patterns can be understood. For example, as
shown from Figure 6A,B, the value of where the cycles
emerge depends on parameters such as d. If
d is decreased, the oscillations emerge for higher values of . Consequently, when an episode is started, the system first jumps toward a high stable steady state, then with increased this high state becomes unstable and oscillations emerge. Thus, our
model when the slow variable is included may exhibit, depending on
parameter values, episodes that have phases of near steady high-activity before oscillations start, as sometimes seen
experimentally (Fig. 1, bottom panel).
Figure 7B shows the variations of oscillation period with
. For the lowest possible value of allowing oscillations, the period is finite. The period lengthens as increases, first slowly and then abruptly just before the oscillations disappear (it becomes infinite at the exact point where the low oscillatory level meets the
intermediate, unstable state). Except for the very end of the range,
the cycle period ranges between 5 and 10 units of time (approximately 7 for the activity shown on Fig. 6C). Therefore, because the
typical cycle period measured experimentally is ~1 sec for embryonic
day (E) 10 embryos, d = 2 predicts that the time
constant of the depression variable d is of the order of 200-400 msec. This is comparable to time scales reported in other systems (Chance et al., 1998).
Slow activity-dependent depression leads to
episodic behavior
In the chick embryo, after an episode of activity, evoked
monosynaptic and polysynaptic potentials are depressed and recover with
a time constant of ~1 min (Fedirchuk et al., 1999). In addition, after an episode, the membrane potential of ventrally located spinal
neurons hyperpolarizes and subsequently recovers as a depolarizing ramp
until the next episode. Collectively, these observations indicate the
presence of a slow (time scale minutes) activity-dependent form of
depression that regulates network excitability and has been proposed to
control the occurrence of episodes (O'Donovan and Chub, 1997).
Although the mechanisms for these changes are unknown, they could occur
through a modulation of cellular properties or a prolonged "slow"
form of synaptic depression. In the following, we formulate and analyze
models for each of these two possibilities.
The two-variable system presented in the previous section models a
bistable, recurrent excitatory network that can exhibit persistent
states of low and high activity, the latter being oscillatory. In
contrast, the chick's spinal cord shows spontaneous alternation between two such states. Here, we add a slow, third variable that enables the model to generate sustained, repetitive episodes of cycling
at high activity separated by low-activity phases, thereby mimicking
the spontaneous, episodic rhythm of the cord. We show that the
mathematical structure that underlies the episodic behavior is the same
for both types of slow activity-dependent depression. Therefore, the
two models behave in similar ways, say, when perturbed by a brief
stimulation or noisy input, and they are affected similarly by some
parameter variations. In a later section, however, we show that the
models lead to different predictions for variations in connectivity and
therefore the models can be distinguished experimentally.
Slow modulation of cellular excitability: -model
Intracellular recordings have shown that there is a slow ramp
depolarization in motoneurons of chick embryonic spinal cord (N. Chub
and M. J. O'Donovan, unpublished observation), suggesting that after
an episode the functional threshold () of neurons is raised and
slowly decreases until the next episode. We can therefore imagine that
increases during an episode and decreases during the silent phase.
This means that the growth and decay of is activity-dependent,
analogous to a slow adaptation mechanism. During high activity the
threshold rises, eventually terminating the activity, and then it
recovers during the interepisode quiet phase. When the threshold
decreases adequately, the network's small amount of spontaneous
activity can trigger another episode. We formulate below a kinetic
model for that yields this behavior.
From our dynamical systems viewpoint and the previous sections, we can
predict the model system's behavior as follows (we will confirm this
below in Figure 8 with a simulation).
First, because will move very slowly, we expect that the
a d subsystem will evolve relatively rapidly to one
of its two attractors (Fig. 7A) and then track it as moves. Suppose the network is in the low-activity state, as after an
episode. Under this condition will slowly decrease, and the
a d subsystem will slowly track the Z-curve's lower
branch until the left knee is reached. The system then rapidly moves to
the oscillatory high-activity state, starting an episode. During the
episode now increases, and as indicated by the thick gray
curves on Figure 7A, the cycle amplitude also gradually
increases. Moreover, the cycle period increases too (Fig.
7B), but only slightly at the beginning of the episode, then
significantly near the end of the episode. Note, this significant drop
in period just before episode termination is a generic feature of this
type of model and is typical of many experiments in chick cord (Fig. 1,
top panel). Eventually, reaches the critical
level c where activity can only jump back to the lower
steady state, terminating the episode. As then decreases again
(recovery), the process will be repeated. Thus, the presence of
bistability of the a d subsystem for a range of values suggests a mechanism for the slow occurrence of episodes, in the
same way that the bistability of the simple recurrent network over a
range of connectivity values suggested a mechanism for the generation
of cycles during an episode. Our prediction of the slow rhythm's
a trajectory rests on the assumptions
(i.e., on our construction) that (1) the fast a d
subsystem is bistable, (2) evolves very slowly, and (3) evolves
in the proper directions during the active and quiescent phases.
These features form the essence of the fast/slow dissection technique
for analyzing multiple time scale oscillations and excitability of this
sort (Rinzel and Ermentrout, 1998).
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Figure 8.
A, Time courses of a and for
the -model. During an episode increases, until the episode
stops. B, Corresponding trajectory in the
(a, ) plane, superimposed on the bifurcation diagram.
Also plotted is the -nullcline (dot-dashed curve). Below
this nullcline, the trajectory flows leftward ( < 0, recovery during the silent phase); above it the trajectory flows
rightward ( > 0, depression during an episode). The
points numbered 1, 2, 3, and
4 correspond to the same states of the network on both
A and B. During an episode, the trajectory passes
sequentially 1-2-3-4; during a silent phase it goes from
4 to 1.
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As for a and d, we model the variations of with a simple, first-order kinetics:
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with an increasing (sigmoidal) function of
a and  being two to three orders of
magnitude greater than a. Thus will increase from
low values if activity is high and decrease from high values if
activity is low.
The time courses of a and for the full three-variable
model resulting from the interaction of the a d fast
subsystem and the slow subsystem are shown in Figure
8A. As expected, the behavior is episodic, with rhythmic
cycling during episodes. The system's trajectory actually lives in the
three-dimensional phase space, a d . In
Figure 8B we project the trajectory onto the (a ) plane, enabling us to see the dynamic behavior along with
the static bifurcation diagram from Figure 7A. We see
qualitatively what we anticipated in the preceding paragraphs, with covering the bistable range back and forth. Although the trajectory
does not exactly follow the bifurcation diagram, it would if we had made larger.
It is important to emphasize that both onset and termination of the
episodes are controlled by the value of , as shown from the
trajectory in the (a ) plane. Also shown in
Figure 8B is the -nullcline, the set of points for which
= 0, that is, = (a). Below this nullcline, < 0, therefore is decreasing, and conversely, for any point above the
nullcline is increasing. Thus, decreases during the silent
phase and increases during an episode. Obviously, the position of the
-nullcline relative to the a d bifurcation
diagram is critical for the existence and timing of episodic
rhythmicity (see below).
This graphical representation allows one to easily predict/understand
the effect of parameter variations on the episodic rhythm. Suppose the
cellular adaptation is recruited more easily (say,
is decreased). This lowers the -nullcline. If it drops too much it
will intersect the Z-curve's lower branch. At this intersection, all
variables, including , are at a stable steady state of low activity;
the network will remain silent. On the other hand, if the -nullcline
is too high, can stop increasing before the end of the episode, and
the network might enter continuous oscillation. Therefore, the position
of the nullcline, or in other words, the value of a for
which the average cellular excitability in the network starts to get
depressed, has to be within a certain range for episodic oscillations
to occur.
The episode durations and silent phase lengths depend on two factors.
First, they depend on the range of values that covers between the beginning and end of the episodes, that is, the horizontal distance between points 4 and 1 on Figure
8B. Increasing this range dilates both the episode and
silent phase in the same way, because the paths 1-2-3-4
and 4-1 are increased accordingly. As illustrated below,
this range can be changed by varying any single parameter of the fast
subsystem. Second, the episode durations and silent phase lengths
depend on the velocity with which varies. Because this
velocity is inversely proportional to , an increase
in this time constant increases similarly the time it takes to depress
the network and the recovery period, so both duration and silent phase
change by the same factor. However, this velocity is also proportional
to the horizontal distance between the current point in the
a plane and the -nullcline, because
= (a) . Therefore, if the nullcline is changed [that is, the
function (a) is changed], velocities are
modified. However, this change in the velocity depends on the activity
state (high or low) of the system. For example, if the nullcline is
moved down, the velocity decreases during the silent phase because
(a) is reduced, but increases during the episode because (a)
is increased. Therefore, episode and silent phase lengths may be
affected in opposite ways when a parameter of the slow subsystem is
changed. The distinction between the effects of parameters of the fast
versus slow subsystems will be important subsequently to distinguish
the two models.
Long-term synaptic depression: s-model
As was considered previously, it is also possible to interpret the
slow form of activity-dependent depression as arising synaptically rather than from a slow modification of cellular thresholds. Indeed, this mechanism has been proposed to account for the slow recovery of
synaptic potentials after an episode (Fedirchuk et al., 1999). Calling
s the new, slow variable, the system becomes:
As for d, a high value of s (near 1) means
"not depressed," so s is a decreasing
function of a. Note that the fraction of nondepressed
synapses is now s·d, and the effective connectivity is
n·s·d. The activity obtained with this model is
qualitatively similar to the one obtained with the -model, as shown
in Figure 9. Here again, the onset and
termination of episodes are controlled by the slow depression variable
(s), therefore silent phase and episode durations are
determined in the same way. The reason the -model and the
s-model are so similar is that variations of s induce the same qualitative changes in the dynamics of the fast subsystem as do variations of . Note that because depression is
associated with smaller values of s, in contrast to the
-model where large means larger depression, the
bifurcation diagram of Figure 9B has opposite (left/right)
orientation to that of Figure 8B.
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Figure 9.
A, Time courses of a and
s for the s-model. During an episode s
decreases, until the episode stops. As for the -model (Fig.
7B), the cycle period increases near the end of an episode.
A little after t = 400, a stimulation was applied,
which triggered an episode. This episode was shorter because recovery
of s was not complete. B, Corresponding
trajectory in the (a, s) plane, superimposed with the
bifurcation diagram and the s-nullcline (dot-dashed
curve). C, Recordings from an E10 cord. Stimulating
quickly after a spontaneous episode triggers a short episode.
Black triangles indicate time of stimulation.
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Simulation experiments with the models
To further characterize the models' behaviors and compare them
with experimental data, a few manipulations were conducted. The model
network can be perturbed by a brief stimulation. It is also important
to understand how the model is affected by varying its parameters,
because the parameter variations may be identified with experimental
manipulations or with evolving system conditions that occur during
development. In addition to the comparison with experimental data, the
following simulations offer some predictions that are testable experimentally.
Activity resetting. In the chick spinal cord, it is
easy to trigger an episode during a silent phase with a brief
electrical stimulus. In addition, if the stimulation is applied shortly
after a spontaneous episode, the evoked episode is much shorter than the average spontaneous episode, as illustrated on Figure
9C. This effect was observed on all preparations tested
(>20). It is to be expected if the silent phase is a period of
recovery from depression, and it is indeed a characteristic feature of the models presented here. We have illustrated this point for the
s-model on Figure 9A,B. Again exploiting the
graphical representation of Figure 9B, we see that to evoke
an episode the stimulus must be substantial enough to bring the
activity above the S-curve's middle branch. Otherwise, activity
returns immediately to the low steady state. Because the vertical
distance between the unstable steady state and low-activity state is
highest just after an episode and progressively declines, the smaller
the amount of time after an episode, the harder it is to successfully
evoke a new episode, in agreement with experiments. These results are
also observed with the -model because it relies on the same
mechanism of episode generation.
Changing a parameter of the fast subsystem. The main
effect of changing one parameter of the fast a d
subsystem is to change the values of the slow variable at the beginning
and/or end of episodes (or, equivalently, at the end and beginning of
silent phases) by deforming the shape of the bifurcation diagram and, in many cases, without moving it much relative to the slow variable's nullcline. Therefore, as mentioned above (section describing the -model), silent phase and episode duration are affected in the same
way, because the slow variable runs through the same range during the
episode and the silent phase.
Figure 10 illustrates the effects of
changing parameters in the fast subsystem: the steepness of
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ABSTRACT
INTRODUCTION
