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The Journal of Neuroscience, May 15, 1998, 18(10):3831-3842
Intersegmental Coordination of Limb Movements during Locomotion:
Mathematical Models Predict Circuits That Drive Swimmeret Beating
Frances K.
Skinner1 and
Brian
Mulloney2
1 Playfair Neuroscience Unit, The Toronto Hospital
Research Institute, and Institute of Biomedical Engineering, University
of Toronto, Toronto, Ontario M5T 2S8, Canada, and 2 Section
of Neurobiology, Physiology, and Behavior, University of California,
Davis, CA 95616-8755
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ABSTRACT |
Normal locomotion in arthropods and vertebrates is a complex
behavior, and the neural mechanisms that coordinate their limbs during
locomotion at different speeds are unknown. The neural modules that
drive cyclic movements of swimmerets respond to changes in excitation
by changing the period of the motor pattern. As period changes,
however, both intersegmental phase differences and the relative
durations of bursts of impulses in different sets of motor neurons are
preserved. To investigate these phenomena, we constructed a cellular
model of the local pattern-generating circuit that drives each
swimmeret. We then constructed alternative intersegmental circuits that
might coordinate these local circuits. The structures of both the model
of the local circuit and the alternative models of the coordinating
circuit were based on and constrained by previous experimental results
on pattern-generating neurons and coordinating interneurons.
To evaluate the relative merits of these alternatives, we compared
their dynamics with the performance of the real circuit when the level
of excitation was changed. Many of the alternative coordinating
circuits failed. One coordinating circuit, however, did effectively
match the performance of the real system as period changed from 1 to
3.2 Hz. With this coordinating circuit, both the intersegmental phase
differences and the relative durations of activity within each of the
local modules fell within the ranges characteristic of the normal motor
pattern and did not change significantly as period changed. These
results predict a mechanism of coordination and a pattern of
intersegmental connections in the CNS that is amenable to experimental
test.
Key words:
pattern generation; coordination; interneuron; constant
phase; excitation; mathematical model; frequency changes
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INTRODUCTION |
Effective locomotion using limbs
located on different segments of an animal's body requires that the
nervous system coordinate firing of different populations of motor
neurons that innervate these different limbs. Limbs on different
segments are normally coordinated so that they move through repeated
cycles of protraction and retraction with a particular phase relative
to their neighbors, a phase that is preserved when the periods of these
movements change. In both terrestrial vertebrates and arthropods, the
CNS can produce the fully coordinated motor output needed to drive these movements without any sensory input, but how this performance is
accomplished is not understood.
Swimmerets are paired limbs found on four or five abdominal segments
that propel a crayfish through the water when it swims forward. When
they are active, the left and right swimmerets of each segment move in
phase through cycles of power strokes and return strokes, and
swimmerets on neighboring segments differ in phase by ~25%. The most
posterior pair starts each cycle of movements. These periodic movements
are driven by a complex motor pattern (Fig.
1) that can be produced by the isolated
abdominal nervous system (Hughes and Wiersma, 1960 ; Ikeda and Wiersma,
1964). When the system is uniformly excited, neither intersegmental
phases nor relative durations of bursts of impulses in motor neurons change significantly as period changes (Mulloney et al., 1993; Braun
and Mulloney, 1995; Mulloney, 1997). Moreover, any neighboring pair of
segments, isolated from the rest, can produce normally coordinated
swimmeret activity (Paul and Mulloney, 1986). How does the nervous
system do it?
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Figure 1.
A, Motor pattern that drives one
swimmeret when the swimmeret system is active. Bursts of impulses occur
alternately in axons of power stroke (PS3) and return
stroke (RS3) motor neurons. B,
Coordinated output to four swimmerets on different segments.
PS2 PS5, Recordings from the power stroke branch of the
swimmeret nerves in abdominal segments 2-5. Lat.,
Latency; Dur., duration; Per., period;
Rel. Dur., relative duration.
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In a previous paper (Skinner et al., 1997), we put aside the cellular
details of the swimmeret system to consider it as a chain of four
segmental oscillators that were coupled bidirectionally to their
nearest neighbors. Given certain values of its parameters, this
phase-coupled oscillator (PCO) model could reproduce the changes in
period and intersegmental phase caused by uniform and nonuniform
excitation in the system (Braun and Mulloney, 1995). These simulations
predicted that (1) excitation must affect the properties of each
segmental oscillator; (2) coupling between segments is asymmetric, but
ascending and descending coupling are about equally strong, and either
ascending or descending coupling alone can cause a phase difference of
~25%, with posterior leading; and (3) changes in intersegmental
coupling do not affect period significantly. What do these properties
mean in neural terms, and how do they emerge from the cellular
components and synaptic organization of the swimmeret system?
In this paper, we introduce a cellular model of the local
pattern-generating module that incorporates properties of its known components. We then construct alternative circuits that intersegmental coordinating interneurons might make to link neighboring modules and
compare the dynamics of these alternatives with the performance of the
swimmeret system itself. From this comparison, one circuit emerges with
dynamics that are similar to those of the real system. These results
illustrate in cellular terms the asymmetry required by the PCO model,
illustrate mechanisms that cause complex properties to emerge, and
predict a pattern of intersegmental connections in the CNS.
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MATERIALS AND METHODS |
Physiological experiments
Crayfish, Pacifastacus leniusculus, were obtained
from commercial suppliers and maintained in aerated tanks of tap water
at 12-15°C. Crayfish were anesthetized on ice before dissection. The
recording methods have been described in detail elsewhere (Sherff and
Mulloney, 1996; Mulloney, 1997). The level of excitation was controlled
by superfusing preparations with different concentrations of carbachol
(Mulloney, 1997).
Each cycle of activity in the swimmeret motor pattern is defined as
beginning with a burst of impulses in power stroke motor neurons in the
most posterior ganglion, A5 (Mulloney and Hall, 1987). The definitions
of bursts of impulses, their periods, durations, and phases are
illustrated in Figure 1. Relative durations are defined as the duration
of each burst divided by the period of the cycle in which it occurred
(Fig. 1). To extend these terms to computations of nonspiking neurons,
we defined the "burst" of each nonspiking cell as starting when its
membrane potential rose above its threshold for transmitter release and
lasting as long as its potential remained above this threshold (see
below).
Computational procedures
Local interneurons and graded synapses. We modeled
each local interneuron as a cell with a single compartment (Perkel and Mulloney, 1978; Edwards and Mulloney, 1984, 1987; Mulloney and Perkel,
1988). Each cell was constructed with two voltage-gated currents, an
inward "Ca" current that activated rapidly relative to an outward
"K" current that activated more slowly, and as many synaptic
currents as were needed to construct the circuit under consideration
(Skinner et al., 1994; Sharp et al., 1996). The first-order ordinary
differential equations that describe these cells and their currents
(Morris and Lecar, 1981) have been used before to study neurons in
synaptic circuits (Skinner et al., 1994; Sharp et al., 1996). The
equations that describe each nonspiking local interneuron and its
synapses are:
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(1)
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(2)
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(3)
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where
where C is the capacitance of the cell; V
is its membrane voltage; t is time;
Iext is any imposed current;
gL, gCa,
and gK are the maximal leak, calcium, and
potassium conductances, respectively; VL,
VCa, and VK are
the reversal potentials of the leak, calcium, and potassium currents,
respectively; M and
N are the fractions of open calcium and
potassium channels at steady state, respectively; N is the
fraction of potassium channels that are actually open;
 N is the rate constant of potassium channel opening, and
 N is the minimum of N; V1
and V3 are the voltages at which half the
channels are open at steady state; and V2 and
V4 are voltages with reciprocals that are the
slopes of the voltage dependence of Ca and K channel opening at steady
state.
Each synapse onto the cell controls a synaptic current that is a term
in Equation 1. gsyn is the maximal synaptic
conductance; S is the instantaneous synaptic activation;
S is the steady-state synaptic activation;
Vsyn is the reversal potential of the synaptic current; Vpre and Vthresh
are the membrane potential and the synaptic threshold potentials of the
presynaptic cell, respectively; and Vslope is
the voltage with a reciprocal that is the slope of the voltage
dependence of synaptic conductance. The term (1  S) S occurs in Equation 3 because
the time constant, S, refers only to decay of synaptic
current, not to a combination of rise times and decay.
To calibrate the model, we used measured membrane time constants, input
resistances, and membrane potentials of swimmeret motor neurons (Sherff
and Mulloney, 1997) and local interneurons (B. Mulloney, unpublished
results).
Intersegmental synaptic connections. The local circuits
within neighboring modules are coupled by a separate circuit of spiking interneurons that conduct bursts of impulses between ganglia (Wiersma and Hughes, 1961; Stein, 1971). Pairs of these coordinating
interneurons originate in each ganglion and fire in phase with the
modules of the ganglion (Paul and Mulloney, 1986), but their targets in neighboring modules are unknown. To connect cells in neighboring modules, we used a "spike-mediated transmission (SMT) threshold" method to simulate these intersegmental interneurons. Accordingly, as
long as the membrane potential of a cell that drives an intersegmental interneuron in one module is above a specified threshold, "spikes" of a specified duration are conducted to the target cells at a specified frequency. Each of these spikes causes a new current in the
target neuron, Iinter.syn, which becomes an
additional term in Equation 1 for the target cell. This current is
described by:
where r is the synaptic activation, and
ginter.syn is the maximal intersegmental
synaptic conductance. If the presynaptic neuron is below the SMT
threshold, r is zero; otherwise it follows a first-order
kinetic model of receptor binding (Destexhe et al., 1994):
During a spike t0 < t < t1,
During an interspike interval (t > t1),
where
t0 is the time when each spike begins;
t1 is the time when it ends;  and  are the
forward and backward rate constants for transmitter binding,
respectively; and T is the transmitter concentration.
These differential equations were integrated numerically on a Pentium
computer with Linux using LSODE (Hindmarsh, 1983), a double-precision
subroutine capable of handling stiff systems of ordinary differential
equations. Except where otherwise specified, the values of parameters
used in these computations are given in Table 4. Multistable behavior
was observed in several cases. However, for all runs performed,
different sets of initial conditions were used to uncover different
stable states and to approximate how large the stability region
was.
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RESULTS |
Each swimmeret in crayfish is driven by its own pattern-generating
module (Murchison et al., 1993) that includes power stroke and return
stroke motor neurons, sensory neurons from the swimmeret, and a set of
unilateral nonspiking local interneurons that generate the alternating
drive to the different motor neurons (Paul and Mulloney, 1985a,b;
Killian and Page, 1992a,b; Sherff and Mulloney, 1997). Each segmental
ganglion has two of these modules that drive the alternating power and
return stroke movements of the two swimmerets of that segment. There
are no significant differences in the intrinsic periods of modules in
different segments (Mulloney, 1997). Within the crayfish nervous
system, certain interneurons have been identified that contribute to
the performance of the swimmeret system and must be considered in any
attempt to construct a mechanistic explanation of that performance.
Unilateral nonspiking local interneurons
The core of the pattern-generating circuit within each module is
the synaptic organization of the nonspiking local interneurons that
drive the motor neurons (Heitler and Pearson, 1980; Paul and Mulloney,
1985a,b; Sherff and Mulloney, 1996). The present evidence indicates
that three types of unilateral nonspiking local interneurons that occur
in each module are required for its normal operation (Table
1). These interneurons have unilateral
structures; that is, their major processes are restricted to one
lateral neuropil (Skinner, 1985). Perturbations of any of these
interneurons resets the timing of ongoing swimmeret activity, and each
interneuron can drive subpopulations of motor neurons effectively. Each
of these interneurons also has distinctive connections within its own
module. Other local interneurons also occur in each segment, but they
do not appear to be necessary components of the local pattern-generating circuit (Paul and Mulloney, 1985a,b).
A model of the local pattern-generating circuit
A minimal local pattern-generating circuit (Fig.
2A) consistent with these data includes
four nonspiking interneurons: two identical 2A interneurons that excite
power stroke excitor (PSE) motor neurons, and two other interneurons,
1A and 1B, with different synaptic input that excite return stroke
excitor (RSE) motor neurons. Other local interneurons are omitted. The
motor neurons export the activity of these pattern-generating neurons
to the muscles of the swimmeret, but their own synaptic interactions do
not play a major role in generating the fundamental alternation of
power stroke and return stroke activity (Sherff and Mulloney, 1996) and
so were not included in these simulations. To compare the activity of
the simulated circuit with the output of the real system, we assume in
this paper that the PSE motor neurons fire whenever and as long as
interneuron 2A is depolarized, and that the RSE motor neurons fire
whenever and as long as interneurons 1A or 1B are depolarized.
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Figure 2.
A, Diagram of the proposed model of
the pattern-generating circuit formed by nonspiking local interneurons
within each module and the connections these interneurons make with
swimmeret motor neurons and coordinating interneurons. 2A, 1A,
1B, Individual local cells. PSE, RSE, Power stroke
excitor and return stroke excitor motor neurons, respectively;
ASC, DSC, ascending and descending coordinating
interneurons, respectively. Gray circles, Graded chemical
inhibitory synapses; gray triangles, graded excitatory
chemical synapses; diode symbol, rectifying electrical
synapse (Paul and Mulloney, 1985a). B, Two series of
computed oscillations of membrane potentials in 2A and
1A cells at 1 Hz (N = 0.003 msec 1) and 3 Hz (N = 0.010 msec1). C, Plots of durations and
relative durations of computed oscillations in cells within the same
module as functions of frequency.
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These are nonspiking neurons, so they are particularly well described
by equations of the Morris-Lecar type. To help the reader distinguish
between interneurons found in crayfish and our models of these
interneurons, we will refer to real interneurons in regular type but
will refer to models as cells named in bold type. Each interneuron was
modeled as a single-compartment cell that had one inward voltage-gated
current, one outward voltage-gated current, a leak current, and as many
synaptic currents as needed (see Materials and Methods). Because the 1A
and 1B interneurons differ in the PSPs they receive during swimmeret
activity (Table 1), we modeled each of them separately as cells
1A and 1B. Because the known properties of the 2A
interneurons are the same, we combined them and their connections into
one cell called 2A.
In our simulation of this circuit (Fig. 2A), the
2A cell and 1A and 1B cells are
functional antagonists that reciprocally inhibit one another.
Individual graded synapses between these neurons were simulated as
postsynaptic membrane conductances, the values of which are functions
of the membrane potential of the presynaptic cell (Skinner et al.,
1994). Individual cells did not oscillate on their own but required
reciprocal inhibition to generate oscillations of membrane potential.
The dynamics of this model resembled those of the real circuit in that
the membrane potentials of each cell oscillated with the same period;
depolarizations of 2A alternated with depolarizations of
1A and 1B (Fig. 2B), and each
of these depolarizations lasted approximately half the period of the
oscillation (Fig. 2C). These features did not change
significantly despite changes in the period of these oscillations (see
below).
Intersegmental coordinating neurons
These interneurons fire in phase with either power stroke (PS) or
return stroke (RS) activity in their home module and project axons from
one ganglion to the next (Table 2). In two pioneering studies, Wiersma and Hughes (1961) and Stein (1969, 1971) described three types of "fibers" in the interganglionic connectives that normally carried phasic information between ganglia whenever the swimmerets were active. Although these coordinating neurons do project
farther than the next ganglion (Table 2), information from more distant
ganglia is not required to produce normal coordination (Paul and
Mulloney, 1986). From this evidence, a minimal set of intersegmental
coordinating neurons that couple two neighboring modules would be two
ascending neurons that fire in power stroke phase, and one descending
neuron that fired in return stroke phase. In the models described here,
simulated coordinating axons connected only to targets in neighboring
segments.
In this series of simulations, we did not explicitly simulate each
coordinating interneuron (see Materials and Methods) but instead
assumed that coordinating interneurons that fire during bursts of
impulses in PSE motor neurons are driven by synaptic input from cell
2A, and those that fire with RSE motor neurons are driven by
synaptic input from cells 1A or 1B (Fig.
2A). To simulate their firing, each ascending and
descending unit generated periodic impulses of specified duration and
frequency as long as the local cell that drove it was depolarized above a specified threshold. We simulated the postsynaptic consequences of
these impulses as synaptic currents in putative target neurons in
neighboring modules (see Materials and Methods). Each of these events
triggered a change in a synaptic conductance in the target neuron.
Possible structures of the circuit that coordinates
neighboring modules
There are at least two parallel ascending axons that conduct
information to unknown targets in the next anterior module (Table 2).
These axons fire bursts of impulses late in the PS phase of the cycle
of their home ganglion, when the 2A interneurons are depolarized. In
our models, there were three possible target neurons, and each of the
connections could be excitatory or inhibitory. To discover circuits
made of these components that performed like the real system, we did a
series of simulations of particular alternative circuits. Because
normal coordination can be produced with any two neighboring ganglia
(Paul and Mulloney, 1986), we made two models of the local
pattern-generating circuit (Fig. 2A, local cells)
with the same values for their parameters (see Table 4) to simulate the
two swimmeret modules on one side of the abdomen. Because we were
concerned with intersegmental coordination, not bilateral coordination,
we constructed one local circuit for each segment.
The PCO model predicted that ascending connections alone should be able
to produce stable phase differences of ~25%, so we began by
considering only ascending connections. There are 12 possible circuits
with two ascending connections to three possible targets that can be
excitatory or inhibitory. Because in the model the intrinsic properties
of 1A and 1B are the same, a circuit with an
excitatory synapse from 2An + 1 to
1An and an inhibitory synapse from
2An + 1 to 2An
would produce the same phase relationships as an excitatory synapse
from 2An + 1 to
1Bn and an inhibitory synapse from
2An + 1 to 2An
and so forth. This reduces the number of different possible circuits to
7. Therefore, we constructed seven qualitatively different patterns of
ascending connections between the two local circuits (Fig.
3) and explored the properties of each
alternative through a 0.01-0.05 mS/cm2 range of
synaptic conductances in their target cells (Table
3).
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Figure 3.
Diagrams of seven alternative circuits
(1-7) that might be made by ascending
coordinating interneurons with targets in the pattern-generating module
of the next anterior ganglion. Each connection from segment
An + 1 to segment An is driven by
input from cell 2A in the more posterior module (Fig.
2A).
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Table 3.
Intersegmental phase lags of 2A cells in anterior modules
coupled by pairs of ascending intersegmental connections that form
seven alternative circuits (Fig. 3)
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Within each module cells 1A and 1B are antiphase
to cell 2A, so one might expect that an "intermediate" phase of 25% would require the two ascending connections to
"contradict" each other, that is, to be excitatory both to
2A and 1A/1B, or inhibitory both to
2A and to 1A/1B, or excitatory to
1A (or 1B) but inhibitory to 1B (or
1A). These patterns occur in Figure 3, circuits 2, 3, and 5. However, Table 3 shows that only circuits 3 and 5 produced appropriate lags, and circuit 5 was much more robust.
Therefore, the alternative that produced appropriate phase lags for the
widest range of synaptic strengths had an excitatory synapse from
2An + 1 to 1Bn
and an inhibitory synapse from
2An + 1 to 1An
(Fig. 4). Because in each simulation the cellular properties of 1A and 1B were identical, this is equivalent to an excitatory synapse from
2An + 1 to 1An
and an inhibitory synapse from
2An + 1 to 1Bn.
We note that circuit 5 is consistent with the bursts of discrete PSPs
recorded from these 1A and 1B interneurons during normal activity
(Table 1).
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Figure 4.
Diagram of the proposed model of the full
intersegmental circuit that coordinates swimmerets in neighboring
abdominal ganglia, An and An + 1.
1A, 1B, 2A, Individual nonspiking local cells within each
module. The connections between modules are thought to be disynaptic;
each coordinating interneuron is driven directly by a local interneuron
in the module where it originates. Black circles,
Intersegmental inhibitory chemical synapses; white
triangles, intersegmental excitatory chemical synapses; gray
circles, graded chemical inhibitory synapses.
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There is also at least one descending axon that fires bursts of
impulses during the RS phase in the cycle of its home ganglion, when
the 1A and 1B interneurons are depolarized, and conducts this
information to the next posterior module. The PCO model predicted that
descending connections should also be able to produce stable phase
differences of ~25%. However, with only one descending connection, it seemed unlikely that stable intermediate phase lags could occur, and
we found that none of the alternative patterns of descending connections with single cells in the target modules could by themselves produce appropriate lags throughout the same range of postsynaptic conductances. If we postulate two descending connections, then just as
for the ascending coupling, there are seven possibilities to
investigate. Like the ascending alternatives (Table 3), only three of
the seven alternatives might be expected to produce intermediate phase
lags. Moreover, the PCO model predicts that ascending and descending
coupling would be asymmetric, which eliminated the pattern of
connections in ascending circuit 5, leaving us with two alternatives.
We found that if the descending axon inhibited both 1A and
2A in the next posterior module (Table 4), the model circuit produced
appropriate lags. No discrete PSPs have been recorded from 2A
interneurons (Paul and Mulloney, 1985b).
A cellular model of the intersegmental coordinating circuit
This analysis of the dynamics of alternative ascending and
descending patterns of connections leads us to propose a model of the
intersegmental circuit that incorporates a minimal number of the known
components of the system organized in a specific asymmetric pattern.
For modules positioned in the interior of the series, the full
coordinating circuit is illustrated in Figure 4; modules located at the
anterior or posterior ends do not get input from the anterior or
posterior direction, respectively. In this coordinating circuit, either
ascending connections or descending connections alone can coordinate
neighboring modules with a phase difference of ~25%.
Modulating the period of the model
To test the full model of four local modules connected by this
circuit (Fig. 4), we first established a set of standard values for
cellular and synaptic parameters (Table 4) that allowed the system to
oscillate at ~2 Hz with intersegmental phases in the range of
20-25%, criteria that fall in the middle of the normal operating
range of the swimmeret system (Mulloney et al., 1993). Next we
investigated the effect on period of changing various parameters in the
model (see Table 4). To do so, we examined an individual module to
determine which parameters controlled the period of its oscillations
and then extended this examination to the fully interconnected system
of four modules. Systematically changing
Vthresh, S,
gsyn, ginter.syn,
gCa, gK,
gL, V1,
V2, V3, and
Iext affected period, but either the range of
possible periods was less than twofold, or relative duration and
intersegmental phases were not preserved. However, we discovered that
modulating the rate constant of outward currents,
N, by changes in N or V4 could cause more than a twofold change in
period, and that the relative durations of depolarizations were
preserved despite these changes (see Figs. 2B,C, for
the solitary module, 6B), so we used changes in
N to simulate changes in the level of
excitation of the system (Mulloney, 1997). Hence, as predicted by the
PCO model, we found that the frequency of coordinated oscillations of
the system was determined largely by properties of its individual modules, not by properties of the intersegmental connections.
Performance of the cellular model
To assess the performance of this model, we varied the frequency
of its oscillations from 1 to 3.2 Hz by altering the rate of activation
of the outward current in individual cells (Fig. 2B)
and compared the intersegmental phases and the relative durations of
depolarizations of cells in each module with the intersegmental phases
and durations of bursts in PSE and RSE motor neurons recorded from
experimental preparations active at different frequencies.
In experimental preparations that expressed swimmeret motor patterns
with different periods, intersegmental phases of bursts in PSE motor
neurons showed some individual variation (Fig.
5A), but the mean values of
the phase of each segment were independent of burst frequency. In
these preparations (Mulloney, 1997), PSE bursts in A2, A3, A4, and A5
were recorded at different burst frequencies. In this figure, the mean
phases of PSE bursts in A2, A3, and A4 relative to PSE bursts in A5
were calculated for four different ranges of burst frequencies, and the
points plotted are means of all experiments within that range.
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Figure 5.
Comparison of intersegmental phases of PSE motor
neurons measured in experimental preparations (A)
with phases of depolarizations of 2A cells computed in the
model (B). Mean phases of bursts of impulses recorded
from PSE axons in A2, A3, A4, and A5 were calculated for four ranges of
cycle frequencies. The plotted points are means of all experiments
within that range. At the highest frequency, the experimental point for
A3 is offset horizontally to make the error bars unambiguous. In the
computational analysis, we calculated phases of periodic
depolarizations of 2A cells at different frequencies of
oscillations. These frequencies were changed by modulating N
between 0.003 and 0.010 msec1.
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In our simulations, there were no components corresponding directly to
PSE motor neurons, but we had assumed that bursts of impulses in PSE
motor neurons were driven by 2A when these cells were
depolarized (Fig. 2A), so we compared the phases of 2A depolarizations in different modules computed at
different frequencies (Fig. 5B). The phases of 2A
depolarizations in different modules, relative to depolarizations of
2A in A5, were similar to the phases of bursts of impulses
in PSE motor neurons in different segments relative to bursts in PSE motor neurons in A5.
A distinctive feature of the normal motor output of the swimmeret
system is that the relative durations of bursts of impulses in PSE and
RSE motor neurons are preserved as the period of motor pattern changes.
To accomplish this preservation, the durations of each burst decrease
as period decreases (Braun and Mulloney, 1995). If this compensation
were perfect, the relative duration of each burst would be constant
despite changes in period. To compare the performance of the swimmeret
system with the performance of the model, we calculated the mean
relative durations of bursts of impulses in PSE motor neurons recorded
in preparations expressing coordinated motor patterns with different
cycle frequencies (Fig. 6A) and compared them
with the relative durations of depolarizations of 2A cells
computed in simulations with different cycle frequencies (Fig.
6B). At frequencies <2.4 Hz, the relative durations of the system and the model are not different. At higher frequencies, the computed relative durations of 2A are briefer than the
measured relative durations in PSE motor neurons.
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Figure 6.
A, Relative durations of bursts in PSE motor
neurons recorded in experimental preparations at different cycle
frequencies. The mean relative durations of bursts of impulses recorded
from PSE axons in A2, A3, A4, and A5 were calculated for four ranges of
cycle frequencies. The plotted points are means of all experiments
within that range. In this figure, these points are offset about
the mean frequency to make the error bars readable. B,
Relative durations of depolarizations of 2A cells as
functions of frequency. Frequency modulation was accomplished with the
method described in Figure 5.
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Sensitivity of the model to synaptic parameters
In this model, the kinetics of intersegmental synapses were much
faster than those of the graded synapses between local cells within
each module. This difference proved to be important if the model was to
approximate both the phase differences and the relative durations of
the real system. If local synaptic time constants were too slow, e.g.,
1.0 sec, then intersegmental factors could dominate the behavior of the
local module such that depolarizations of local cells were terminated
prematurely by incoming inhibitory signals. If local synaptic time
constants were too fast, e.g., 0.25 sec, then the behavior of the
individual module quickly stabilized and prevented intersegmental
signals from significantly affecting phase lags. We found that a time
constant of 0.5 sec for the graded local synapses gave an appropriate
balance between local and intersegmental factors (Table 4). This
balance of synaptic parameters might be linked to our observations
(above) that changes in either N or
V4, both intrinsic properties of
individual cells, changed the frequency of the system but preserved
phases and relative durations. Previous modeling of reciprocal
inhibitory circuits found that changes in intrinsic properties could
cause either increases or decreases in the period of oscillations of
the circuit, depending on the underlying mechanisms that drive these
oscillations (Skinner et al., 1994; Sharp et al., 1996). These
mechanisms could be "intrinsic" or "synaptic" (Skinner et al.,
1994). Flexibility in the mechanism generating the oscillations of each
module might be a way to preserve intersegmental phase and relative
duration as frequency changes.
Contributions of ascending connections and descending connections
to the stability of the performance of the model
Given that either the ascending connections or descending
connections of the model alone could cause phase lags in the right range when the local circuits were oscillating at ~2 Hz, we examined how ascending and descending connections performed at different frequencies. We ran a series of simulations with either only ascending or only descending connections and different intrinsic cycle
frequencies. Both the ascending-only and the descending-only circuits
produced coordinated activity with substantial differences in
intersegmental phase, posterior segments leading, at all frequencies
tested (Fig. 7A). The
performance of these circuits differed most at low frequencies; ascending-only output had small intersegmental phase differences, but
descending-only output had larger phase differences. This suggests that
the better performance of the full circuit (Fig. 5) results from
balancing contributions of ascending and descending connections.
View larger version (25K):
[in this window]
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|
Figure 7.
Features of the coordination of a four-module
chain computed when only ascending connections or only descending
connections of the proposed coordinating circuit (Fig. 4) were present.
A, Phases of depolarizations of 2A cells in
anterior modules as functions of cycle frequency. B,
Relative durations of depolarizations of cells 1A,
1B, and 2A in module A4 computed at different
frequencies. Frequency modulation was accomplished with the method
described in Figure 5.
|
|
The relative durations of depolarizations of the three local cells in
each module, in contrast to their intersegmental phases, were similar
in the ascending-only and descending-only series (Fig. 7B)
and similar to their relative durations computed with the full circuit
(Fig. 6). Results from module A4 are plotted in Figure 7B;
the relative durations of the homologous cells in other modules were
similar to these.
Other simulations indicate that the full model, with both ascending and
descending connections (Fig. 4), is more resilient to perturbation of
these intersegmental synapses than is either the ascending-only or
descending-only circuit. In particular, we found that the full circuit
could maintain normal intersegmental coordination through a larger
range of intersegmental synaptic strengths than could the
ascending-only circuit. For smaller changes, both circuits maintained
coordination.
| |
DISCUSSION |
Coordination of limb movements during normal locomotion is
fundamental to behavior, but the mechanisms that achieve it are little
understood. Although phasic sensory feedback is critical if these
movements are to be adapted to local requirements and to accidents of
individual histories (Wilson, 1968; Grillner, 1981), the CNSs of most
animals are organized so that essential features of this coordination
can be accomplished without any sensory input. The swimmeret system in
crayfish illustrates this property particularly clearly (Hughes and
Wiersma, 1960; Ikeda and Wiersma, 1964). We present a cellular model of
the swimmeret coordinating circuit that is based on experimental
results and predictions made by our PCO model of this system (Skinner
et al., 1997) and that incorporates essential features of the
architectonics of the system.
This model effectively reproduces key features of the performance of
the swimmeret system: invariant intersegmental phases and relative
durations of activity. Many of the alternative circuits that we
simulated did not perform at all like the real system (Table 3), so it
is remarkable that one alternative was so effective. Only when the four
local modules were linked by this pattern of coordinating connections
were both the intersegmental phases of their oscillations and the
relative durations of depolarization of individual local cells
preserved as the cycle frequency of the system changed (Figs. 5, 6).
For the model to maintain normal intersegmental phase differences, the
strengths of the ascending and descending signals had to be similar,
and the time constants of local and intersegmental synaptic connections
had to be balanced, but within these constraints, either ascending or
descending axons connected in this way would cause an ~25% phase
difference (Fig. 7) with posterior leading. The model was constructed
by assuming a reciprocal inhibitory organization of particular local
interneurons (Fig. 2A) and by using experimental data
(Tables 1, 2) to restrict the number and properties of intersegmental
connections. Thus, this model illustrates how local interneurons and
intersegmental interneurons could be organized in a particular
configuration to coordinate swimmeret movements.
One widely applicable result of this study is how changes in
excitation were modeled. Although the swimmeret system can be modulated
by different putative transmitters (Mulloney et al., 1987; Sherff and
Mulloney, 1991; Braun and Mulloney, 1993; Chrachri and Neil; 1993), and
differential modulation of particular phases of the pattern is known
(Mulloney et al., 1997), the cellular mechanisms by which frequencies
of periodic motor patterns change are largely unknown. While studying
the dynamics of the model, we found it difficult to vary its frequency
widely. One common tactic for changing firing frequencies of model
neurons is to change injected current (Skinner et al., 1993). However,
in this model, intersegmental coordination and relative durations were not maintained through a range of cycle frequencies if steady currents
were injected into individual cells. Instead, relative durations
increased as the frequency of oscillations increased, unlike the
response of the swimmeret system to changes in excitation (Fig. 6). We
tried varying other cellular parameters (Table 4), but only changing
the kinetics of the outward current in Equation 1 allowed the circuit
to match the performance of the system. Modulation of outward currents
exerts substantial control over period and phase in other
pattern-generating circuits (Marder and Calabrese, 1996), and despite
the simplifications inherent in our simulations, our results suggest
that modulation of potassium currents might be part of the mechanism by
which the period of swimmeret beating is controlled.
Simplifications inherent in these models
The proposed model of the swimmeret coordinating circuit
incorporates significant cellular details of the swimmeret system, but
in simplified ways. Both the models of individual cells and of their
local and intersegmental synapses are attempts to capture only their
essential features. The model of each local pattern-generating circuit
(Fig. 2A) is a minimal cellular model, and some of
its details have not yet been examined either computationally or
experimentally. In constructing the four local circuits that simulated
the modules in four neighboring segments, we further simplified this
proposed local circuit by including only the cells that represented the nonspiking local interneurons (Table 1) but omitting both sensory components and motor neurons. In particular, the omission of motor neurons and the assumption that depolarization of 2A is synonymous with firing of PSE motor neurons should be tested
experimentally.
In the model of this coordinating circuit (Fig. 4), we included a
minimal set of elements that simulate the known coordinating interneurons (Table 2). The method used simplified these interneurons extremely; they played no role in their home segment; their firing frequencies were fixed and did not change when the frequencies of the
local circuits that drove them changed. In physiological experiments,
coordinating interneurons do respond to changing levels of excitation
(H. Namba and B. Mulloney, unpublished results). Preliminary
simulations in which their spike frequencies increased uniformly as
excitation increased gave intersegmental phases and relative durations
in the same ranges as those reported here. These simplifications might
cause the imperfect correspondence between model and experiment,
especially at the higher frequencies. This feature may become more
important when we consider nonuniform excitation (Braun and Mulloney,
1995).
Relation of the PCO model to this model
It is productive to consider a complex system from different
perspectives but sometimes difficult to map the results of different levels of consideration. In an earlier study (Skinner et al., 1997), we
considered only the phase differences between segments and asked what
properties of coordinating information linking neighboring segments
were necessary to produce phase-differences of 25%. Predictions from
that study, that either ascending coupling or descending coupling alone
would produce the appropriate phase difference and that ascending and
descending coupling were asymmetric, suggested the strategy that let us
sort efficiently through different circuits that interneurons might
make (Fig. 3). PCO models as a class, because they include only phase
and coupling as parameters, are limited in the extent to which they can
illuminate the mechanisms that determine the performance of the system.
They may predict asymmetry but are silent on the nature of that
asymmetry (Pearce and Friesen, 1988; Williams, 1992b; Sigvardt, 1993).
This cellular model, in contrast, illustrates how a coordinating
circuit can be asymmetric yet equally effective in ascending and
descending directions. The "coupling" between modules is asymmetric
in that the targets of ascending connections are different from the
targets of descending connections and also in that the signals these
targets received are not the same sign.
Comparison of this model with similar models of circuits
coordinating locomotion in other animals
The spinal control of swimming in leeches, lamprey, and tadpoles
has also been simulated with cellular models of segmental oscillators
(Williams, 1992a; Grillner et al., 1995; Roberts et al., 1995; Jung et
al., 1996). The complexity of neuronal properties included in these
models differs widely; our descriptions of the nonspiking local neurons
in each module included voltage-gated currents and graded postsynaptic
currents integrated in a single compartment and so are of intermediate
complexity. A detailed model of the segmental and intersegmental
circuits that control swimming in leeches has been developed and
refined by interaction of simulation and experiment (Friesen, 1989;
Friesen and Pearce, 1993). This model differs from the one presented
here in that its connections are justified by physiological recordings,
and that the contributions of cellular dynamics to the performance of
the circuit have not yet been investigated (cf. Canavier et al., 1997).
Walking in terrestrial vertebrates is driven by spinal pattern
generators, one for each limb, or perhaps one for each joint in the
limb, that are coordinated by interneurons in the spinal cord
(Rossignol et al., 1993; Berkowitz and Stein, 1994). The properties of
motor neurons and their modulators (Kiehn et al., 1997) and the
contributions of segmental circuits to limb movements (Stein et al.,
1995) have been described, but no cellular model of these coordinating
circuits has been proposed. Walking in insects and crustaceans involves
more pairs of limbs, but smaller numbers of neurons, than it does in
vertebrates. Each walking leg has its own module (Ronacher, 1991). The
cellular organization of reflexes in the walking legs (Burrows, 1992;
Wolf and Laurent, 1994) and properties of intersegmental interneurons
that might coordinate legs during walking have been well characterized
(Laurent, 1988; Laurent and Burrows, 1989a,b). Cholinergic reagents
elicit coordinated stepping motor patterns from isolated thoracic
ganglia of locusts (Ryckebusch and Laurent, 1994) and crayfish
(Chrachri and Clarac, 1990), but the coordination of the pattern
generators for walking in these isolated ganglia has proven more
variable than those produced by the isolated swimmeret system (Sillar
et al., 1987; Ryckebusch and Laurent, 1994; Berkowitz and Laurent, 1996a,b), and the models of the coordinating circuit for walking that
have been proposed are still quite abstract (Müller and Cruse,
1991; Beer et al., 1992; Ryckebusch et al., 1994).
Our results suggest specific features both of the structure of each
module and of the intersegmental coordinating circuit. Given the
assumption that the core of each module is the reciprocal inhibitory
connections made by specific nonspiking local interneurons (Fig.
2A), they predict that neighboring modules are linked
by a specific configuration of intersegmental connections (Fig. 4). The
model illustrates that coordinating interneurons do not need to affect
pattern-generating neurons in their home modules directly; instead,
their axons conduct impulses to particular targets in neighboring
modules. Many of these details are in principle testable by
physiological experiment.
| |
FOOTNOTES |
Received Nov. 12, 1997; revised Feb. 20, 1998; accepted Feb 25, 1998.
This work was supported by a University of California president's
fellowship to F.K.S. and by National Science Foundation Grant BNS
95-14889 to B.M. We thank Nancy Kopell for provocative discussions,
Karen Sigvardt, Ken Britten, Hisaaki Namba, and Wendy M. Hall for
reading this manuscript critically, and Wendy M. Hall for data analysis
and preparing illustrations.
Correspondence should be addressed to Dr. Frances K. Skinner, Playfair
Neuroscience Unit, The Toronto Hospital, Western Division, 399 Bathurst
Street, MP12-303, Toronto, Ontario M5T 2S8, Canada.
| |
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February 1, 2004;
91(2):
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[Abstract]
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Y. Manor, A. Bose,, V. Booth, and F. Nadim,
Contribution of Synaptic Depression to Phase Maintenance in a Model Rhythmic Network
J Neurophysiol,
November 1, 2003;
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A. A.V. Hill, M. A. Masino, and R. L. Calabrese
Intersegmental Coordination of Rhythmic Motor Patterns
J Neurophysiol,
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B. Mulloney
During Fictive Locomotion, Graded Synaptic Currents Drive Bursts of Impulses in Swimmeret Motor Neurons
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July 2, 2003;
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S. R. Jones, B. Mulloney, T. J. Kaper, and N. Kopell
Coordination of Cellular Pattern-Generating Circuits that Control Limb Movements: The Sources of Stable Differences in Intersegmental Phases
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April 15, 2003;
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J. Cang and W. O. Friesen
Model for Intersegmental Coordination of Leech Swimming: Central and Sensory Mechanisms
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M. A. Masino and R. L. Calabrese
A Functional Asymmetry in the Leech Heartbeat Timing Network Is Revealed by Driving the Network across Various Cycle Periods
J. Neurosci.,
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A. A. V. Hill, M. A. Masino, and R. L. Calabrese
Model of Intersegmental Coordination in the Leech Heartbeat Neuronal Network
J Neurophysiol,
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M. A. Masino and R. L. Calabrese
Period Differences Between Segmental Oscillators Produce Intersegmental Phase Differences in the Leech Heartbeat Timing Network
J Neurophysiol,
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N. Tschuluun, W. M. Hall, and B. Mulloney
Limb Movements during Locomotion: Tests of a Model of an Intersegmental Coordinating Circuit
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W. O. Friesen and C. G. Hocker
Functional Analyses of the Leech Swim Oscillator
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H. Namba and B. Mulloney
Coordination of Limb Movements: Three Types of Intersegmental Interneurons in the Swimmeret System and Their Responses to Changes in Excitation
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Top
Abstract
Introduction
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