1Committee on Neurobiology and 2Department of Neurology, The University of Chicago, Chicago, Illinois 60637
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ABSTRACT |
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Foss, Jennifer and
John Milton.
Multistability in Recurrent Neural Loops Arising From Delay.
J. Neurophysiol. 84: 975-985, 2000.
The dynamics of a recurrent inhibitory neural loop composed of a
periodically spiking Aplysia motoneuron reciprocally connected to
a computer are investigated as a function of the time delay, , for
propagation around the loop. It is shown that for certain choices of
, multiple qualitatively different neural spike trains co-exist. A
mathematical model is constructed for the dynamics of this
pulsed-coupled recurrent loop in which all parameters are readily
measured experimentally: the phase resetting curve of the neuron for a
given simulated postsynaptic current and
. For choices of the
parameters for which multiple spiking patterns co-exist in the
experimental paradigm, the model exhibits multistability. Numerical
simulations suggest that qualitatively similar results will occur if
the motoneuron is replaced by several other types of neurons and that
once
becomes sufficiently long, multistability will be the
dominant form of dynamical behavior. These observations suggest that
great care must be taken in determining the etiology of qualitative
changes in neural spiking patterns, particularly when propagation times
around polysynaptic loops are long.
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INTRODUCTION |
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A large body of experimental and
theoretical evidence indicates that abrupt and lasting changes in
neural spiking patterns can be produced in certain neurons
(Canavier et al. 1993, 1994
; Fitzhugh
1969
; Guttman et al. 1980
;
Hounsgaard et al. 1988
; Lechner et al.
1996
), simple neural circuits, such as central pattern generators (CPGs) (Canavier et al. 1999
; Harmon
1964
; Hopfield 1984
; Kleinfeld et al.
1990
), and large neural populations (Kelso 1995
;
Kelso et al. 1992
; Krüse and Stadler
1995
; Milton 2000
; Wilson and Cowan
1972
) by applying a brief perturbation, such as an electrical
pulse. This phenomenon is referred to as multistability; each
qualitatively different spiking pattern as an attractor. In contrast to
the effect of neuromodulators that change neural spiking patterns
through their effect on membrane properties such as ion conductances
(Benson and Levitan 1983
), in a multistable situation
changes in spike patterns occur because the effect of the perturbations
is to cause a switch between the co-existent attractors. Multistability
has been associated with a large variety of cortical functions
including memory (Barnes et al. 1997
;
Cowan 1972
; Foss et al. 1996
;
Mensour and Longtin 1995
; Zipser et al. 1993
), visual perception (Krüse and Stadler
1995
), and motor control (Eurich and Milton
1996
). Moreover, it has been suggested that it might be
possible to develop therapeutic strategies based on the manipulation of
multistability to treat diseases such as epilepsy (Milton
2000
).
Multistability also arises in time-delayed feedback mechanisms
(an der Heiden and Mackey 1982; Foss et al. 1996
,
1997
; Ikeda and Matsumoto 1987
). In particular,
multiple oscillatory attractors have been observed in experiments
involving laser optical devices (Aida and Davis
1992
; van Tartwijk and Agarwal 1998
) and
electronic circuits (Foss et al. 1997
; Losson et
al. 1993
). Time delays are an intrinsic property of the nervous
system and arise because axonal conduction times and distances between
neurons are finite. The propagation time through a polysynaptic loop is
even further prolonged by the effects of synaptic mechanisms and neural
integration times (Milton 1996
). However, relatively
little attention has been given to the possibility of
time-delay-related multistability by the neurobiological community with
the notable exceptions of modeling studies (Foss et al. 1996
,
1997
; Mackey and an der Heiden 1984
;
Plant 1981
). An essential requirement for multistability to arise in a delayed feedback loop is that the time delay be longer
than an intrinsic time scale of the element experiencing feedback, such
as the period of an oscillation (Foss et al. 1996
, 1997
)
or a response time (Ikeda and Matsumoto 1987
; van
Tartwijk and Agrawal 1998
). In the CNS, axonal conduction times
(Miller 1994
; Nunez 1995
; Waxman
and Bennett 1972
) and propagation times through polysynaptic
loops (Eurich and Milton 1996
; Iijima et al.
1996
; Miles et al. 1988
; Milton
and Longtin 1990
; Villa and Abeles 1990
;
Wright and Sergejew 1991
) approach several hundreds of
milliseconds. Since neural spike frequencies can exceed 10 Hz
(Abeles et al. 1993
; Rapp et al. 1985
;
Richmond and Optican 1990
; Zipser et al.
1993
), i.e., inter-spike intervals can be less than 100 ms, we
anticipate that multistability might occur.
Here we investigate the neural spiking patterns that are produced by
the delayed neural feedback loop shown in Fig.
1A. This circuit is
constructed from a neural oscillator that is subjected to delayed
feedback. The neural oscillator may be a single neuron, a CPG, or a
neural population; the feedback loop may represent, for example,
respectively, a recurrent inhibitory loop in the olfactory cortex or
hippocampus, peripheral sensory feedback to a CPG, or feedback from
brain stem and sub-cortical structures to an epileptic cortical region.
These feedback loops are pulse coupled, i.e., the coupling is
characterized by a pulse-like interaction between the neural oscillator
and the loop. This pulse coupling is a direct consequence of the fact
that when neurons are physically separated, interactions between them
are in the form of discrete synaptic potentials (PSPs) driven by
spikes. The critical parameter for determining the dynamics of a
pulse-coupled neural network is spike timing (see, for example,
Judd and Aihara 1993). Consequently all of the
physiological processes involved in the transmission of information
around the neural loop can be incorporated into a single number, the
time delay. To implement this loop experimentally, we developed the
recurrently clamped neuron paradigm shown in Fig. 1B: the
membrane potential of a spiking Aplysia motoneuron is
monitored by a computer. Each time the motoneuron spikes, the computer,
after a time delay
, injects an inhibitory current pulse (ICP) to
mimic inhibitory postsynaptic currents (IPSCs), which would be caused
by natural neural feedback. Although single cells have been found to
exhibit multistability, the type of multistability that we study here
results entirely from the effect of delay on the feedback loop. As will
become apparent, the advantage of studying this feedback loop is that
the dynamics can be predicted from experimentally measured parameters.
Thus it is possible to directly compare theory and observation.
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Our observations are presented in three parts. First, we demonstrate
that qualitatively different neural spiking patterns co-exist in the
recurrently clamped Aplysia neuron shown in Fig. 1B. This multistability occurs primarily when the time
delay, , in the feedback loop is longer than the intrinsic period,
T, of the spiking Aplysia neuron. Second we
demonstrate that the multistability that occurs in the recurrently
clamped neuron can be understood in terms of a mathematical model that
incorporates two experimentally measurable parameters:
and the
phase resetting properties of the neuron. Finally, based on this model
and known phase resetting properties, we predict that the dynamics that occur when the Aplysia motoneuron in the feedback loop is
replaced by bursting and beating neurons (Dror et al.
1999
; Schindler et al. 1997
) will be
qualitatively similar. Moreover, numerical simulations show that in all
of these cases, as
becomes sufficiently long, multistability
becomes the dominant behavior expected for neural oscillators subjected
to pulse-coupled delayed feedback.
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METHODS |
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Slowly adapting Aplysia motoneurons
Slowly adapting buccal motoneurons of Aplysia
californica were used. Aplysia care, immobilization,
and dissection were carried out as described in Church and Lloyd
(1994). Unless otherwise stated, experiments were performed in
artificial seawater with high divalent cations (33 mM
Ca2+, 165 mM Mg2+), which
raises the firing threshold, thereby reducing spontaneous activity in
the ganglion. Typical input resistances were ~2-5 M
.
Recurrent neural clamping device
The design of the recurrent neural clamping device is similar in
spirit to that previously used to "dynamically" clamp neurons (Sharp et al. 1993a,b
) and crayfish stretch receptors
(Diez Martinez and Segundo 1983
; Kohn et al.
1981
; Vibert et al. 1979
), except that a time
delay,
, is explicitly inserted.
Periodic spiking was induced by injecting a depolarizing DC current step (2-6 s) with an AxoClamp 2B amplifier in bridge mode. The smallest DC current (usually 1-20 nA) that caused the neuron to fire repeatedly (inter-spike intervals 20-200 ms) and reliably was used for all subsequent experiments. Under these conditions, the neurons fired several spikes before reaching their highest firing frequency which then gradually declined over 2-5 s. All experiments were started when the change in firing rate was smallest and conducted during the following slow adaptation period.
The membrane potential is monitored through the recording electrode of
the AxoClamp. An analog signal proportional to the membrane potential
is passed to a Pentium 66 computer via an A/D board (AD 2210, Real Time
Devices). The computer detects neural spikes and when feedback is
running, each time the computer detects a spike, at a time after,
it triggers the AxoClamp to inject a single square hyperpolarizing
current pulse (ICP;
5 to
25 nA) into the neuron, thereby closing
the feedback loop. This feedback is allowed to run for lengths of
25-75 times the normal firing period. For the first
amount of time
that the feedback is on, there will ordinarily be no pulses because of
the delay. During this time, it is possible to simulate various initial
conditions of the system by adding a patterned pulse sequence of length
. In this way it is possible to identify the co-existing
qualitatively different spiking patterns.
Phase resetting
Neurons were induced to fire periodically using the same conditions as during the feedback experiments. Phase resetting curves were measured by injecting a single ICP and measuring the effect on the resulting inter-spike interval (see RESULTS). Because of the variation in the neuron's period from one trial to the next and because the firing rate of the neuron accommodates within a run, each phase resetting and feedback experiment had a control run before it to establish the average spiking frequency of the neuron at that time.
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RESULTS |
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Multistability
Figure 2, A and
B, shows typical spike trains generated by a recurrently
clamped Aplysia neuron when is greater than its normal
firing period. Both patterns were observed for the same choice of ICP
and delay. They differ in the position of the ICPs and in the spiking
frequency. In Fig. 2C, a brief electrical pulse causes a
change from the pattern in B to the one in A.
After the new pattern is established, a second electrical pulse causes
the system to go back to the spiking pattern seen in Fig.
2B. Herein we refer to each stable pattern of neural spiking
as an attractor. The observations in Fig. 2, A-C,
demonstrate that there is multistability, i.e., at least two attractors
are present.
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Figure 2D shows the same neuron and , but the timing of
the second added pulse is different. In this case no second switch in
spiking pattern occurs. This illustrates the finding that the time at
which the electrical pulse is added is critically important for causing
switches between attractors. Previous studies have shown that randomly
timed pulses have a low probability of causing switches between
attractors in the bursting R15 neuron of Aplysia (Lechner et al. 1996
).
Model
We developed a mathematical model for the recurrently clamped
neuron with the goals of determining the number attractors which co-exist as a function of and the timing of electrical pulses to
cause switches between the co-existent attractors. Since our focus is
pulse-coupled neural oscillators, our model comes from the perspective
of the phase resetting properties of neural oscillators. The advantage
of this approach is that the phase resetting properties of a neural
oscillator can be readily measured experimentally. We first discuss the
phase resetting properties of a regularly spiking Aplysia
buccal motoneuron and then present the model (see also
APPENDIX).
Phase resetting properties
The effect of a stimulus on spike timing of a periodically firing
neuron has extensively been studied both experimentally (Perkel
et al. 1964; Pinsker 1977a
,b
; Schindler
et al. 1997
) and theoretically (Best 1979
;
Glass et al. 1984
; Winfree 1980
). The effect of a perturbation on the dynamics is summarized by the phase
resetting curve (PRC) (Glass and Mackey 1988
;
Glass et al. 1984
; Winfree 1980
). The PRC
relates the phase at which the perturbation arrives to the subsequent
phase shift in the neuron's spike times.
Figure 3A shows the effect of
a single ICP on spike timing of a periodically spiking
Aplysia buccal motoneuron. In this experiment the
unperturbed period, T, was 97.2 ms. The phase, , at which the electrical pulse was delivered is
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(1) |
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Following the arrival of the ICP, the timing of subsequent neural
spikes is shifted from the time that they would have appeared had there
been no ICP (compare - - - to in Fig. 3A). We say that
the phase of the oscillator has been reset by an amount
. In our
model for the dynamics of the recurrent neural loop, the phase reset is
used to predict the time of the very next spike. The phase reset,
,
can be calculated from the time of the next spike
t1, i.e.,
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(2) |
There are several conventions for expressing the phase resetting curve.
We have chosen one used frequently in the literature which is a plot of
the new phase, ', versus the old phase,
(Best
1979
; Glass and Mackey 1988
; Guevara et
al. 1981
; Winfree 1980
). The PRC is measured by
introducing ICPs at different phases,
, and measuring the phase
reset,
, which results. The new phase,
', can be easily
calculated from
by using the relation
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(3) |
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Recurrent loop dynamics
To understand the dynamics of the delayed recurrent loop, it is
necessary to keep track of the times that the spikes and the ICPs
occur. To illustrate the procedure, a schematic representation of a
spiking pattern is shown in Fig. 5. Each
time the neuron spikes (indicated by - - -), an ICP is delivered at
time later (indicated by the horizontal solid line). We denote
n as the phase of the
nth ICP, due to a spike Sn. Although it is possible to first
calculate the time that the nth ICP is delivered and then
determine
n, it is much more efficient and straight-forward to perform all calculations in the phase domain. The
phase at which the ICP is delivered depends on the number and timing of
any ICPs that occurred in the interval between
Sn and
n. Except when
spikes and ICPs occur, phase increases continuously with time at a rate
T
1. Since the
time that has elapsed between Sn and
n is
,
n is
equal to
/T less some amount that reflects the net amount
of reset from ICPs and spikes. For the example
interval shown by
the horizontal bar in Fig. 5, proceeding left to right,
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(4) |
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(5) |
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(6) |
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The numbers N and k depend on the previous phases
n
1,
n
2,
etc. A bookkeeping shortcut allows the number of ICPs and inter-spike
intervals which occur in the delay interval to be easily accounted for:
We introduce a new variable,
, defined as
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Using this definition of , we can rewrite Eq. 6 as
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(7) |
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(8) |
The basis for multistability in a delayed recurrent loop can be readily
appreciated by determining the fixed points, *, of Eq. 7
for any given value of
. The fixed points, i.e., when all
n =
*, are the solutions of the equation
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(9) |
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Figure 6 illustrates this approach for the case of a recurrent
inhibitory loop involving an Aplysia neuron. When
/T < 1 (Fig. 6A), having more than one
fixed point, and hence multistability, becomes very unlikely unless
there is positive resetting (see DISCUSSION). Having more
than one fixed point becomes more likely when
/T > 1 (Fig. 6, B and C). In the case of this
Aplysia neuron and many other living and model neurons, the
stability condition can be satisfied for more than one critical point
for given ranges of
(this communication). Thus it becomes possible
to observe multistability experimentally in delayed recurrent loops
once the delay becomes sufficiently long. It should be noted that some neurons and other types of limit cycle oscillators can give rise to
behaviors more complicated than multiple stable co-existent fixed
points (see, for example, Foss et al. 1997
).
Recurrent loop spiking patterns
We used two different strategies to compare the prediction of
Eq. 7 to the experimentally measured dynamics of the delayed recurrent loop. First, we determined whether the same initial condition
leads to the same neural spiking pattern in the experiment and the
model. When two or more spiking patterns co-exist, there must be sets
of initial conditions that produce one or the other spiking pattern. In
the recurrently clamped neuron, the initial conditions correspond to
neural spike patterns of length . In the experiment, for the first
seconds that the feedback is turned on, there are no ICPs (the
earliest time that an ICP can occur is
). During this time it is
possible to use the computer to insert a given pulse pattern of length
, i.e., to initialize the system. Figure
7 shows membrane potential traces from an
Aplysia buccal motoneuron in two consecutive delayed
feedback experiments. The delay and all other experimental parameters
are the same in both experiments. The first dashed vertical line
indicates the time at which feedback was turned on and the time
interval between this, and the second dashed vertical line is the
"initial condition." In Fig. 7A, the initial condition
corresponds to
n
1 = 2.79 and
n
2 = 2.79 (see
Fig. 7 legend for how this was determined). Since k = 2 (Eq. 8),
n
3 does not
affect the computation. Thus we can use Eq. 7 to compute
n = 2.68. Continuing in this fashion, we
numerically obtain the steady state value of
* = 2.74. The
experimental steady-state
value is 2.80, obtained from averaging
the
s of several of the ICPs after the system has reached an
equilibrium. In Fig. 7B,
n
1 = 2.20 and
n
2 = 2.20.
n given by Eq. 7 is 3.33. Iterating repeatedly, we find a steady-state value of
* = 3.23. The
experimental
value in this case is 3.27. This exemplifies how the
map can be used to predict which attractor will be reached using a
given initial condition. It follows from these observations that
Eq. 7 can be used to calculate the time(s) at which an ICP
should be introduced to cause a switch between attractors.
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The patterns shown in Fig. 7 are both fixed points of the map in
Eq. 7. The multistability depends on the notion that the delay time can span different numbers of inter-spike intervals (different N values). In Fig. 7A spans two
full intervals plus a fraction, while in Fig. 7B it spans
three intervals and a fraction (see horizontal bar). This is possible
because ICPs fall at different phases in the two patterns, causing
different phase resets and thus different sized intervals.
Second we measured the spiking patterns generated by the delayed
recurrent loop as a function of and compared these to the predictions of Eq. 7. In particular, we compared the
values at which the ICPs occur as a function of
. Figure
8A,
, gives the prediction
using Eq. 7 with the PRC shown in Fig. 3B for the time delay up to four times the period. In the figure, for each value
of
, several different initial conditions, varying systematically in
frequency, were used to start numerical iteration of the map. For this
PRC, all solutions represent periodic firing patterns, i.e., the firing
patterns differ only by their rate. As can be seen, for
between 2.0 and 2.3 or 2.95 and 3.7, two firing patterns are predicted to co-exist.
The
in Fig. 8A represent the experimental observations
for the same neuron and ICP (examples of 2 of the coexisting neural
spiking frequencies are shown in Figs. 2 and 7). As predicted by the
model, two different neural spiking frequencies exist for ranges of
.
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Although the agreement between observation and prediction shown in Fig.
7A is quite good, it can be seen that Eq. 7
systematically gives lower predictions for the s than is observed
experimentally. In the next section, we examine one possible
explanation for this discrepancy.
Fixed phase resetting curves
The co-existent spike patterns we observed consist of a series of
inter-spike intervals, each of which contains one ICP at the same phase
(see Figs. 2 and 7). It is possible that an ICP arrives when the
spiking neuron is still some distance from its limit cycle trajectory.
This is supported by our observation that wave form and inter-spike
interval are often slightly different for an interval following one
containing an ICP. If this happens, then the PRC will not accurately
predict the new phase. To test this hypothesis, we measured the fixed
phase resetting curve (FPRC) for the spiking neuron. An FPRC is
obtained by repeatedly injecting ICPs at a fixed phase, , and
measuring the reset,
, once a stable situation is attained (Fig.
9A). The FPRC is equivalent to
the recurrently clamped neural loop when
< 1. Figure
9B shows the FPRC for the same spiking neuron shown in Fig.
3. The
is the curve fit through the FPRC. The fact that the PRC and
FPRC are not the same (compare
and - - -) has been reported
previously for other excitable cells (Kunysz et al.
1997
; Lewis et al. 1987
).
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In Fig. 8B, gives the prediction of Eq. 7
using the FPRC shown in Fig. 9B. In Fig. 8B,
are the same experimental points shown in Fig. 8A. Clearly
the agreement between prediction and observation is better when
repeated fixed phase stimuli rather than single stimuli are used to
compute
(
) for the map. These observations indicate that in the
delayed recurrent loop the spiking Aplysia neuron has not
quite returned to its limit cycle attractor before the next ICP arrives.
Other PRCs
Figure 10 shows the PRCs
(left) and the predicted dynamics of a recurrent
pulse-coupled neural loop using the PRCs for a tonically firing
Aplysia buccal motoneuron reset by square hyperpolarizing current pulses (A), model of a bursting R15
Aplysia neuron reset by simulated inhibitory synaptic
currents (Dror et al. 1999) (B), tonically
firing rat cortical neuron reset by square hyperpolarizing current
pulses (Schindler et al. 1997
) (C), and
Morris-Lecar model, a two variable model of a periodically firing cell,
reset by a square hyperpolarizing current pulse (Rinzel and
Ermentrout 1998
) (D).
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The structure of the steady-state phases as a function of are
remarkably similar for all the cases. This is surprising since the PRCs
are quite dissimilar. Once the time delay is greater than the period
for neural spiking (i.e.,
/T > 1), it becomes possible for multiple spiking patterns to co-exist over certain ranges
of
. Regions in which multiple spiking patterns co-exist alternate
with regions in which only one spiking pattern is possible. As
increases, the regions for which multiple spiking patterns co-exist
become larger while those for which only one spiking pattern exists
become smaller. Figure 11 demonstrates
using the recurrently clamped Morris-Lecar model that once
/T > 5, multiple attractors exist for all
's.
Similar results were obtained for the other three types of neurons.
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DISCUSSION |
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In our model of the recurrently clamped neuron, we have made use
of the fact that a neural circuit composed of spatially separated elements must be pulse-coupled. This assumption enables the dynamics of
the recurrent loop to be modeled using the PRC measured from the
isolated spiking neuron. Thus instead of developing a model based on,
for example, detailed descriptions of the various currents across the
neuronal membrane (see, for example, Foss et al. 1996), the model can be based on the experimentally measured PRC. The major
advantage of the recurrently clamped neural loop is that we are able to
quantitatively compare prediction to observation. This is precisely the
type of experimental paradigm we needed to be able to conclusively
demonstrate that multistability could arise in the nervous system
simply because time delays in neural circuits are sufficiently long.
However, our results should not be interpreted as implying that a
recurrent pulse-coupled inhibitory neural loop that includes a limit
cycle oscillator is the only type of delayed neural feedback mechanism
capable of producing multistability. Mathematical and computer studies
suggest that there are a large number of neurobiologically relevant
types of feedback mechanisms which can generate multistability (see for example, an der Heiden and Mackey 1982
; Campbell
et al. 1995
; Eurich and Milton 1996
;
Foss et al. 1996
; Ikeda and Matsumoto 1987
; Milton and Foss 1997
). The demonstration
that such mechanisms are also relevant to the occurrence of
multistability in the living nervous system awaits the development of
appropriate experimental paradigms and methods to evaluate all of the
required parameters.
The tonically firing Aplysia neuron used in our experiment
does not by itself exhibit multistability under our experimental conditions. The multistability observed is due to the time delay in the
loop. Ikeda and Matsumoto (1987) were among the first to draw attention to the observation that multistability could arise in a
delayed feedback mechanism once the time delay became longer than a
critical time scale. This time scale depends on the nature of the
delayed feedback mechanism under consideration. For example, for a
feedback control mechanism of the form
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(10) |
It is not difficult to imagine how the condition that > T might be satisfied in large polysynaptic recurrent neural
loops. In such loops, many factors can contribute to the delay and
consequently the propagation time through the loop may be considerably
longer than would be estimated from the conduction velocities. For
example, the latency of the pupil light reflex is ~300 ms compared
with ~20 ms, which would be estimated from the conduction velocities and the length of the loop (Milton 1996
). However, it is
important to note that it is not the absolute magnitude of
which is
relevant, but rather its value relative to another time scale. Thus if
T is short, the condition for multistability, i.e.,
/T > 1, may be satisfied even in recurrent
inhibitory loops composed of two neurons (Foss et al.
1996
).
The PRCs we measured for the Aplysia buccal motoneurons can
contain regions of phase advance as well as phase delay. The phase advances which sometimes occur when ICPs are given early in the cycle
(i.e., < 0.1) have been suggested to arise because of the
deactivation of a voltage-gated K+ conductance
(Lytton and Sejnowski 1991
); consequently the outward K+ current decreases sufficiently to allow the
cell to reach threshold earlier. The phase delays observed when ICPs
are given later in the cycle reflect the effects of the brief change in
stimulating current on the time required for the membrane to recharge.
For many PRCs, an abrupt change in
' occurs for
between 0.8 and 1.0 (see Fig. 10A). Since the PRC for a limit cycle
oscillator is theoretically predicted to be a continuous function of
(Glass and Mackey 1988
; Glass et al.
1984
; Winfree 1980
), we choose to fit the PRC
with a continuous function (see APPENDIX). For many excitable cells, the change in
' following a change in
occurs so
abruptly that the PRC is often fit with a piecewise defined function
(Graves et al. 1986
; Guevara et al.
1981
). In our experiments, the question of the continuity of
the PRC is not that important because attractors do not take on values
of
in the steep, potentially discontinuous region of the PRC. Other
investigators have found complex dynamics, not including
multistability, when stimuli are repeatedly applied with delay
< T falling in this transition region of the PRC
(Kunsyz et al. 1997
). We were unable to convince ourselves that such behavior exists in our paradigm since the noise
level is quite high.
The PRC is a powerful tool to study the effects of perturbations on
biological oscillators (Winfree 1980). Since the PRC can be measured, predictions based on the PRC are directly applicable to
the interpretation of real world phenomena. For example, the PRC has
been used to understand the etiology of certain cardiac arrhythmias
(Guevara et al. 1981
) and the interaction between the
respiratory oscillator and a mechanical ventilator in an intubated subject (Graves et al. 1986
). Mathematical models for
the nervous system that emphasize the generic properties of PRCs have
been used to study fictive swimming in the lamprey (Ermentrout
and Kopell 1994
; Kopell 1995
), the effects of
periodic inhibitory stimulation of cortical slices (Schindler et
al. 1997
), and the occurrence of multistability in ring circuit
models for CPGs (Canavier et al. 1999
; Dror et
al. 1999
) and to develop models to describe the behavior of
large population of neurons (Hoppensteadt and Izhikevich
1997
). The observations in Fig. 10 indicate that for many
neural oscillators the PRCs can be very different, yet the dynamics of
the recurrent loop remain markedly similar. Thus we conclude that the
occurrence of multistability in a delayed recurrent neural loop is a
robust phenomenon. Indeed we observed multistability in 23 of 26 Aplysia motoneurons that were studied in the recurrently clamped neural loop.
In our experimental paradigm, the timing of the ICPs is precisely
controlled. It is reasonable to question whether spike timing could be
sufficiently precise so that the dynamics we observe could be seen in
the normally functioning nervous system. Each of the attractors that
arise in a multistable delayed recurrent loop has associated with it a
basin of attraction (Foss et al. 1996; Milton and
Foss 1997
). This means that the spiking patterns have a certain
stability to perturbations in the timing of ICPs. Thus the issue of the
precision of spike timing is not crucial. Of course, certain
fluctuations in spike timing can cause switches between basins of
attraction (Foss et al. 1997
). Noise-induced switching
between basins of attraction has been suggested to underlie the
fluctuations observed in human postural sway (Eurich and Milton 1996
) and the changing perceptions of visually ambiguous
figures (Krüse and Stadler 1995
).
Multistability can arise at many different levels within the nervous
system. For example, in motor control multistability can arise at the
level of the -motoneuron (Hounsgaard et al. 1988
) and
at the level of local neuromuscular feedback control loops
(Campbell et al. 1995
) or be a property of integrative
control systems such as the control of human posture (Eurich and
Milton 1996
). Here we have emphasized the occurrence of
multistability in recurrent neural loops. It has been suggested that
every neuron within the CNS can be connected to every other neuron by a
pathway involving just a few synapses (Guyton 1976
;
Watts and Strogatz 1998
). Thus from a macroscopic point
of view the nervous system has a recurrently looped structure.
Currently emphasized recurrent neural loops include the multiple
pathways involved in the control of movement (Beuter et al.
1989
), reciprocal thalamo-cortico loops involved in epileptic
seizures (Gutnick and Prince 1972
) and regulation of
states of arousal (Conteras et al. 1996
), the limbic
nervous system loops related to memory (Barnes et al.
1997
) and epileptic seizures (Schwartzkroin and McIntyre
1997
), and the cortical-basal ganglia-thalamus-brain
stem-cortical loops, which participate in the control of movement and
act as gate keepers for the propagation of epileptic seizures
(Proctor and Gale 1997
). As neural loops become longer,
at some point the propagation time around the loop must exceed the
typical inter-spike interval. Our observations suggest that once this
occurs, multistability becomes probable. The recognition that
multistability can be an emergent property of the integrative
organization of the nervous system has implications ranging from
understanding higher cortical processing to the design of therapeutic
strategies for brain diseases.
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APPENDIX: CURVE FITTING |
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To incorporate (
) into our model, it must be expressed as
a function or a lookup table. The effect of noise and the measurement error made using the raw data for a lookup table problematic. The small
number of data points and their unequal spacing made conventional
smoothing algorithms unusable, and simple linear interpolation created
artifacts in the map results due to the nondifferentiable nature of the
function resulting from such interpolation. Therefore we chose to
fit functions to the experimental data. Phase reset data points
measured from Aplysia motoneurons were empirically fit with
a function formed by the product of a parabola and sigmoid, i.e.
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(A1) |
Given the time constraints of experiments (both the phase resetting
curve and the delayed feedback experiments must be done on the same
neuron) and the noise level in our preparation, it was impractical to
adequately explore the phase resetting values for phases close to 1. Theoretically, PRCs must be continuous everywhere modulo 1, including at = 0 and
= 1. Within the resolution of
our experiments, the PRCs data points measured from Aplysia
motoneurons conform with this theory. However, rather than constrain
our functional fits to have cyclic continuity, we choose the function
which best fits the existing experimental data points, regardless of
continuity at
= 1. The advantage of this approach was that we
are able to precisely determine the nature of the PRC for the portion
that is most relevant to the experimentally measured dynamics. The
limitation of this approach is that because of the limited data
at high phase, we sometimes obtain curve fits for which
's at
= 0 and
= 1 are not equal modulo 1 (see Figs. 3 and
9). Although these curve fits are used in Eq. 7, the map
does not generate fixed points with phases near 1. Thus the
discontinuity of the curve fits at
= 1 does not affect the
accuracy of our map's predictions.
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ACKNOWLEDGMENTS |
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We thank L. Fox and P. Lloyd for advice concerning Aplysia and assistance with experiments; J. D. Hunter for help in measuring the phase resetting curves for natural IPSCs; J. Cowan, L. Glass, P. Mason, and A. T. Winfree for helpful comments in the preparation of this manuscript; and J. Crate for design of our data acquisition system.
This research was supported by grants from the National Institutes of Mental Health and the Brain Research Foundation. J. Foss acknowledges support from the Lucille P. Markey Foundation and the Women's Council of the Brain Research Foundation.
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FOOTNOTES |
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Address for reprint requests: J. G. Milton, Department of Neurology, MC 2030, The University of Chicago, 5841 South Maryland Ave., Chicago, IL 60637 (E-mail: sp1ace{at}ace.bsd.uchicago.edu).
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 5 November 1999; accepted in final form 18 April 2000.
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REFERENCES |
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