1Section of Neurophysiology, Department of Physiology, The Panum Institute, 2200 Copenhagen N; and 2CORE, Niels Bohr Institute, 2100 Copenhagen Ø, Denmark
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ABSTRACT |
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Beierholm, Ulrik, Carsten D. Nielsen, Jesper Ryge, Preben Alstrøm, and Ole Kiehn. Characterization of Reliability of Spike Timing in Spinal Interneurons During Oscillating Inputs. J. Neurophysiol. 86: 1858-1868, 2001. The spike timing in rhythmically active interneurons in the mammalian spinal locomotor network varies from cycle to cycle. We tested the contribution from passive membrane properties to this variable firing pattern, by measuring the reliability of spike timing, P, in interneurons in the isolated neonatal rat spinal cord, using intracellular injection of sinusoidal command currents of different frequencies (0.325-31.25 Hz). P is a measure of the precision of spike timing. In general, P was low at low frequencies and amplitudes (P = 0-0.6; 0-1.875 Hz; 0-30 pA), and high at high frequencies and amplitudes (P = 0.8-1; 3.125-31.25 Hz; 30-200 pA). The exact relationship between P and amplitude was difficult to describe because of the well-known low-pass properties of the membrane, which resulted in amplitude attenuation of high-frequency compared with low-frequency command currents. To formalize the analysis we used a leaky integrate and fire (LIF) model with a noise term added. The LIF model was able to reproduce the experimentally observed properties of P as well as the low-pass properties of the membrane. The LIF model enabled us to use the mathematical theory of nonlinear oscillators to analyze the relationship between amplitude, frequency, and P. This was done by systematically calculating the rotational number, N, defined as the number of spikes divided by the number of periods of the command current, for a large number of frequencies and amplitudes. These calculations led to a phase portrait based on the amplitude of the command current versus the frequency-containing areas [Arnold tongues (ATs)] with the same rotational number. The largest ATs in the phase portrait were those where N was a whole integer, and the largest areas in the ATs were seen for middle to high (>3 Hz) frequencies and middle to high amplitudes (50-120 pA). This corresponded to the amplitude- and frequency-evoked increase in P. The model predicted that P would be high when a cell responded with an integer and constant N. This prediction was confirmed by comparing N and P in real experiments. Fitting the result of the LIF model to the experimental data enabled us to estimate the standard deviation of the internal neuronal noise and to use these data to simulate the relationship between N and P in the model. This simulation demonstrated a good correspondence between the theoretical and experimental values. Our data demonstrate that interneurons can respond with a high reliability of spike timing, but only by combining fast and slow oscillations is it possible to obtain a high reliability of firing during rhythmic locomotor movements. Theoretical analysis of the rotation number provided new insights into the mechanism for obtaining reliable spike timing.
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INTRODUCTION |
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Central neurons reveal discharge
patterns with a high degree of variability (Rieke et al.
1997). In the visual cortex, for example, the number and timing
of action potentials in spike trains varies between repeated
presentations of the same sensory stimulus (Britten et al.
1993
; Snowden et al. 1992
; Tolhurst et
al. 1983
). In addition to time-varying global excitability
changes (Arieli et al. 1996
; Gur et al.
1997
), the variations in spike timing may result from either
intrinsic noise in the spike-generating mechanism or from variable
synaptic activity generated by the same recurrent stimulus
(Azouz and Gray 1999
; Shadlen and Newsome 1998
). Recent in vitro studies have shown that cortical neurons can respond with a high reliability of spike timing when stimulated repeatedly with intracellular time-varying inputs (Mainen and Sejnowski 1995
; Nowak et al. 1997
; Tang
et al. 1997
). The reliability of spiking is, however, highly
dependent on the temporal pattern of the input command. Thus
high-frequency fluctuating events, comparable in frequencies with those
observed in vivo, are transmitted more reliably than low-frequency or
DC events. Although cortical neurons may exhibit variations in action
potential threshold in relation to the membrane fluctuations
(Azouz and Gray 1999
), these data suggest a low
intrinsic noise level in the spike-generating mechanisms and indicate
that the variability in spike timing observed in vivo is mainly due to
variations in the underlying synaptic activity generated by the sensory stimulus.
To what extent these observations generalize to other areas of the
mammalian brain is largely unknown. The primary goal of the present
study is to explore this in the context of a rhythmic motor behavior.
We do this by investigating the reliability of spike timing in spinal
interneurons located in laminae X and VIII and the ventromedial part of
lamina VII of the isolated spinal cord of the newborn rat. These spinal
cord areas contain sufficient neuronal networks for the creation of
locomotor rhythms (Kiehn and Kjaerulff 1998;
Kjaerulff and Kiehn 1996
). Previous studies have shown
that many neurons in these areas are rhythmically active during
transmitter-induced locomotor activity in the neonatal rat
(Kiehn et al. 1996
; MacLean et al. 1995
;
Raastad et al. 1996
-1998
; Tresch
and Kiehn 1999
). However, the spike numbers and spike timing of
interneurons in these regions varies from locomotor cycle to locomotor
cycle. This variability of spike timing could be due to a
cycle-to-cycle variability in the synaptic drive to the rhythmically active cells or due to noise in the transfer of presynaptic activity to
postsynaptic spiking. Here we assess whether part of the observed variability of firing rests in the interneuron response properties. To
investigate this, we injected sinusoidal command currents of different
frequencies and amplitudes and measured the reliability of spike timing
in response to these commands. We show that the reliability of spike
timing can be very high in these interneurons, but that the reliability
depends on frequency and amplitude of the stimulus command. By
comparing our experimental data with modeling data and introducing a
new way of looking at reliability of spike timing, we can make precise
predictions about the presynaptic activity that produces a maximal
reliability of spike timing in these neurons. A high reliability of
spike timing will synchronize activity in large groups of cells leading
to an increased motor output.
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METHODS |
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Dissection
The spinal cord was isolated from newborn Wistar rats, generally 1 day old (range 0-2 day). The rats were anesthetized with ether, quickly decapitated, and eviscerated. The spinal cord was split midsagittally from L2 to L5. Both halves of the cord were pinned in the experimental chamber with the cut surface facing up, and perfused with standard Ringer solution containing (in mM) 128 NaCl, 4.7 KCl, 25 NaHCO3, 1.2 KH2PO2, 1.25 MgSO4, 2.5 CaCl2, and 20 glucose, which was oxygenated with 95% O2-5% CO2, keeping the pH at 7.4.
To prevent spontaneous synaptic activity during recordings, D,L-2-amino-5-phosphonovaleric acid (AP-5, 40 µM) and 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX, 20 µM) and occasionally strychnine (0.3-0.5 µM) and bicuculline (20 µM) were added to the Ringer solution. All experiments were performed at room temperature.
Recordings
Patch electrodes were pulled on a two-stage Nashirige Vertical
Puller from 1.5-mm Borosilicate glass without filaments (Clark Instruments, Pangbourne, UK) to a final resistance of 5-10 M. The
pipette solution contained 128 mM K-gluconate, 10 mM HEPES, 0.1 µM
CaCl2, 0.3 mM GTP, 5 mM Li-ATP, and 1 mM glucose
(pH 7.3). Whole cell current-clamp recordings were performed with an
Axopatch 1D amplifier (Axon Instruments, Foster City, CA) and
controlled from a PC using pCLAMP software (Axon Instruments).
Interneurons were recorded in laminae X, VIII, and the ventromedial
area of lamina VII. The recordings were sampled at 6.25 kHz and
filtered at 1 kHz. The current traces were injected as either a
square pulse or sine waves, repeated 10 times with 1-s intervals. This was done for various frequencies and amplitudes for each cell. To
estimate the time constant of the membrane,
, a zap protocol was run
with amplitude yielding subthreshold membrane potential fluctuations
(see Fig. 3).
Analysis
The reliability, P, of firing has been used as a
quantitative measurement of the neural response to repeated identical
stimuli (Mainen and Sejnowski 1995; Nowak et al.
1997
). Here we have calculated P in a way very
similar to that described by Nowak et al. (1997)
. The
exact spike times from all 10 stimuli in a run were projected onto the
time axis. This axis was divided into 3-ms bins. The number of spikes
in each bin was then used for calculations.
P is the total number of spikes in all bins containing more
than 1 spike, divided by the total number of spikes in all 10 stimuli.
In this way, P is a dimensionless number between 0 and 1, where 0 indicates a random response and 1 a nonrandom and
entrained response. Contrary to Nowak et al. (1997), we
did not include the immediately adjacent bins in our calculations of an
event, unless it contained more than one spike. The bin size we used was chosen to be compatible with the slow firing and relative broad
action potentials of neurons in the neonatal rat spinal cord [the
width at 1/2 the peak amplitude of the action potential is
1.67 ± 0.39 (SE) ms for neonatal motor neurons
(Fulton and Walton 1986
) and 3.2 ± 1.5 ms for the
interneurons located in the ventromedial area of the neonatal spinal
cord (Hochman et al. 1994
)]. All calculations and
generation of stimulus files were done using Origin 6.0 (Microcal) and
IDL (Interactive Data Language version 5.2.1: Research Systems) software.
Model
To examine our experimental findings we used the simple leaky integrate and fire (LIF) neuronal model, with a few modifications. The original model considers a cell consisting of an isolated membrane with resistance, R, and capacitance, C. The current through the membrane is denoted I, and the potential across the membrane, Vmem. When no current is injected, the cell will have a natural resting potential, Vrest.
When a current is injected at time t = 0, the potential
of the cell membrane, relative to the resting potential
V(t)=Vmem - Vrest, can be described according to
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(1) |
When injecting a constant current, I0,
the equation has the solution
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(2) |
To mimic action potential firing in the model, we added a threshold for firing, at a value of the potential Vthres (12 mV above resting membrane potential; average value measured in real neurons). On reaching this Vmem = Vthres at some time t0, the membrane potential momentarily resets to the resting potential V(t0) = 0, and the neuron is considered to have generated an action potential.
Thus the classical LIF model does not explicitly contain the form of the action potentials. To compensate for this, we made a small modification to the LIF model. The modification can be summarized in two steps as follows.
1) We introduced a refractory period where the neuron is unable to create a new action potential. This short time interval, dt, was set to 2 ms.
2) An exponential function,
V2, with a time constant so large
that V2
0 after the time interval
dt was added to the solution. The exponential function was
multiplied with a factor B, chosen to give realistic
amplitude of the spike. We use a value of 90 mV for B and
of 6.25 ms
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(3) |
With these two additions to the LIF model, more realistic spiking was observed. It should be stressed, however, that the additions do not change the quantitative results of the simple LIF model in any way, as long as the average firing frequency does not exceed 100 Hz.
This now gave us a neuron model that responded with a regular firing pattern if a depolarizing current injection was large enough and with increasing firing rate for increasing current.
The LIF model is initially a deterministic model that always gives the
same response to a given stimuli, in contrast to a live neuron. Owing
to the many internal processes governing the behavior of a real cell, a
neuron will fire with some variation. To include this behavior in our
model, we added Gaussian distributed white noise with zero mean and
deviation, n. The noise term was added to the
potential V of the model as a sum of 100 sine waves
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(4) |
The magnitude of this noise, the prefactor C, was found by
comparison with the experimental data (see RESULTS;
Comparison of model and real neurons). Eventually we ended
up using noise Vn with a standard
deviation, n, of 0.9 mV.
As a final implication, we note that rhythmically active interneurons
are subject to an alternating synaptic input, which causes the membrane
potential to oscillate between periods of depolarization and
hyperpolarization (Kiehn et al. 1996; Raastad et
al. 1996
, 1998
.) We therefore introduced
time-varying command currents in our model as opposed to the original
square pulse. The simplest time-varying command for our purpose is the
sine wave current I(t) = A
sin (2
t) of frequency
and amplitude A.
We also choose to change the resetting of the membrane potential at the
time of firing, so that V(t0) = Vthres
A0AR just after firing.
Equation 1 now has the solution
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(5) |
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(6) |
A0 characterizes the well-known
low-pass ability of the cell (see, for example, Carandini et al.
1996), as the denominator attenuates the higher frequencies
(see RESULTS).
This sum of the potentials V1, V2, and Vn, together with the threshold for firing and the 2-ms refractory period (see condition 1 above) now constitute the model.
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RESULTS |
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Reliability of spike timing in lumbar spinal interneurons
The membrane potential of rhythmically active interneurons during
locomotion displays a range of different frequencies. In general, fast
postsynaptic potentials with frequencies ranging from 5 to 30 Hz are
superimposed on slower oscillations with frequencies of approximately
0.3-1 Hz (Kiehn et al. 1996; Raastad et al.
1996
-1998
). As a first attempt to mimic this
synaptic profile and thereby investigate the relationship between input
parameters and reliability of spike timing, we injected sinusoidal
command currents of different amplitudes and frequencies in the absence
of locomotion. We compared these results with the response properties
obtained by using DC command currents. In this section, we will give a
brief account of the general observations made with these two stimulus paradigms.
Figure 1A shows an
intracellular recording from an interneuron (middle panel)
when stimulated with a DC square current pulse (amplitude 75 pA). Even though the cell might seem at first sight to fire at regular
intervals, the raster plot (Fig. 1A, top panel) shows that
the firing pattern in repeated trials was actually irregular with a low
reliability (P = 0.136). Characteristically the first
spikes following stimulus initiation in every run appear very close in
time. This high reliability decreases, however, after only two or three
action potentials, confirming previous observations that neurons
respond with low reliability when stimulated with DC currents
(Mainen and Sejnowski 1995; Nowak et al.
1997
).
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Figure 1B (middle panel) shows the firing pattern of an interneuron stimulated with a sine wave of low current amplitude (57 pA, 3.125 Hz). The cellular responses are not identical in the repeated runs of stimuli, but the raster plot (Fig. 1B, top panel) shows a much better alignment of the firing times (P = 0.331) than for the pure DC stimulus. In this example, the membrane voltage lingered around the threshold for firing, making the cell fire only occasionally at the peak. This gave a relatively low P value. However, when the amplitude of the command sine wave was increased (85 pA, 3.125 Hz; Fig. 1C, bottom and top panels), the cellular responses were almost identical for repeated runs, with high reliability (P = 0.868).
Therefore interneurons in the neonatal rat spinal cord seem to have the capability to entrain to an alternating command, here a simple sine wave current, and to produce a very reliable response. This implies the possibility of precise spike timing when exposed to synaptic currents of certain amplitudes and frequencies.
Measurements were made on 34 interneurons for various frequencies
between 0.625 and 31.25 Hz and current amplitudes between 20 and 135 pA. The time constants of the cell membranes varied between 50 and 100 ms, and the input resistance was usually 500-1,000 M, although a
few cells had resistances as low as 215 M
. These values are similar
to those previously observed in interneurons in the neonatal rat spinal
cord (Kiehn et al. 1996
; Raastad et al.
1998
).
The general observation from these experiments was that the reliability of spike timing in response to slowly alternating command currents in general was lower than with sinusoidal command currents at higher frequencies. Likewise, reliability increased with the amplitude of the command current. Thus when injecting sine wave commands at high frequencies (6.125-18.375 Hz) and amplitudes (80-130 pA), very high reliability could be observed (P > 0.9, 7 neurons). Finally, the reliability was low for neurons exposed to a pure DC current (P < 0.3, 5 neurons).
These general statements are, however, contradicted by the observation that the reliability decreased as the amplitude increased beyond a certain limit. Further increase of the amplitude could then increase the reliability again. This relationship between amplitude and the transient drop in reliability was frequency dependent. Another confounding factor is that the amplitude of the voltage response decreased when the frequency of the command current increased. To investigate these relationships in further detail, we used the LIF model and compared it with the experimentally obtained data.
Spike response in the LIF model
The first set of experiments in the LIF model was to test how well it reproduced the general experimental observations.
Figure 2 shows the intracellular response of the LIF model to a DC square current pulse (Fig. 2A) and sine wave commands (Fig. 2, B and C; using the same frequency and amplitudes as the real cell was stimulated with in Fig. 1). As in spinal interneurons, the DC current (75 pA) gave low reliability of spike timing (P = 0.168). Changing to a low-amplitude (57 pA, 3.125 Hz) sine wave increased the reliability (Fig. 2B; P = 0.425), although the responses were not identical for repeated runs, as in the experimental situation (Fig. 1B). Again, the amplitude was too small to evoke an action potential in each cycle. The noise in the model is responsible for these irregularities. Raising the amplitude (85 pA) of the command current markedly increased the reliability of the response of the neuron model (Fig. 2C; P = 0.892). As in the experimental situation, the alignment of the firing times in the raster plot (Fig. 2C; top trace) shows that the spike timing is reproducible from trial to trial, thereby giving a reliable response.
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Cells act as low-pass filters
Our experiments on spinal interneurons and in the LIF-model using
sine wave command currents with the same amplitude and different frequencies revealed a nonlinear connection between the amplitude of
the command current and the amplitude of the recorded voltage. Thus the
amplitude of the recorded voltage decreased with increasing frequencies
(see Figs. 1 and 2). To investigate the nature of this effect, we
constructed a "zap-command" protocol, which was a sine wave with
discreetly increasing frequency (Fig.
3B). The zap command was
scaled so it evoked subthreshold voltage responses. From Fig.
3A it can be seen that running the zap command in an interneuron evoked a voltage response that decreased in amplitude with
increasing frequency. This low-pass effect to an external stimulus has also been described in other neurons (see Carandini et al. 1996) and was seen in all the 34 interneurons recorded.
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In Fig. 3C (+), the amplitude of the voltage response is
plotted as a function of the zap-command frequency. It is clear from this figure that it is only above a certain frequency, the cutoff frequency, that the low-pass effect becomes prominent. In the model, this can also be deduced directly from Eq. 6
(METHODS), where the two ranges of frequencies (low and
high) and the splitting between them occurs at the cutoff-frequency
= 1/(2
RC). The amplitude of frequencies below
this value are not changed, while the amplitudes of frequencies above
are attenuated.
When the theoretical estimate A0 (see
METHODS, Eq. 6) was superimposed on the
normalized experimental curve (Fig. 3C) obtained from the
zap protocol, a clear correspondence was observed, except for higher
frequencies, where the two curves deviate. The theoretical decrease of
the amplitude with a factor 1 is only
maintained until about 10 Hz, after which the experimental voltage
response amplitude decreases with a slower rate, namely
3/4. This relationship was seen in five of
six interneurons. The reason for this deviation between experiments and
theory is not known. The important point, however, is that there is a
good correspondence between the real neurons and the LIF model.
The prominent low-pass effect at higher frequencies made it difficult to compare the reliability of firing at different frequencies of the command current. We therefore turned to a mathematical analysis of the relationship between frequency, amplitude, and reliability of firing.
Mathematical analysis of the relationship between firing behavior, amplitude, and frequency
The LIF model enables us to use the mathematical theory of
nonlinear oscillators to analyze the firing behavior (see, for example,
Alstrøm et al. 1988, 1990
; Jensen
1998
). Using a pure sine wave as stimulus current further
lightens the task of analyzing the problem. We will start by
considering the LIF model in the absence of noise.
Instead of using P as a measurement of reliability, we now investigate the entrainment between the incoming command (the sine wave) and cell firing. We use the rotation number (N) to describe this relationship. N is defined as N = J/K, where J is the sum of all action potentials in a given trial and K is the number of cycles in the sine wave command. In other words, N defines the number of spikes per cycle of the input command. As we shall see later, there is a clear connection between the degree of entrainment to a command and the reliability of the response.
The rotation numbers for the different values of sine wave frequency
and voltage fluctuations RAA0 in the
LIF model are plotted in Fig. 4. This
phase-plot defines areas with constant rotation numbers. The areas with
integer rotation number (e.g., 1, 2, 3, etc.) will generally be the
largest, but with decreasing area as the rotation number increases. The
noninteger areas (e.g., 1/2, 3/4,
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The good relationship between the spike response in the interneurons and the LIF model allows us to use the plot in Fig. 4 to predict the reliability of the firing of a neuron when exposed to a specific sine wave command. The sizes of the areas, also called Arnold tongues, determine the reliability with which a sine wave command is transmitted. Thus a cell responding with N = 1 (that is 1:1 phase locking) will be more reliable than a cell responding with N = 1/2, as the area of the parameter space (amplitude and frequency) for which the rotation number is consistent is larger for N = 1 than for N = 1/2. That is, it is less susceptible to intrinsic variability in the system. This is because when a neuron is in a position in the phase plot close to the border of two rotation areas, it will fire unreliably, as intrinsic "noise" can move it easily between areas of different rotation numbers.
From these considerations, we conclude that the larger the area in the
phase plot a neuron occupies within an Arnold tongue, the more reliably
it will fire (see DISCUSSION). The largest areas in the
phase plots occur for middle to high ( > 3 Hz) frequencies and
middle to high amplitudes of the command. This corresponds with the
increase in reliability seen in the experiments when frequency and
amplitude were increased. These areas also correspond to areas where
the derivatives of the input commands, dI/dt, are high (the derivative increases with amplitude and frequency).
It should be noted that the lower amplitude limit in the graph is determined by the model parameters. This amplitude limit will shift to lower values if the distance between Vrest and Vthres decreases.
In Fig. 5, we have incorporated the low-pass effect in the phase plot. The areas of individual tongues are compressed and stretched. This is because points in the phase plots above the cutoff frequency will be dislocated to the right, because a higher amplitude is needed to evoke the same response as without the low-pass effect. The consequence of this effect is an upper limit for frequencies that evoke spiking (see Cells act as low-pass filters). In the real experiments, we were able to observe 1:1 phase locking for input commands up to 31.25 Hz. It can also be seen that increasing the frequency for fixed amplitude will make the cell fire with lower rotation number until firing eventually stops.
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Finally, Fig. 4 also explains the experimentally observed frequency-dependent relationship between amplitude and the transient drop in reliability (see Reliability of spike timing in lumbar spinal interneurons). In Fig. 4 it can be seen that increasing the amplitude or frequency of input command does not necessarily increase the reliability. Thus going from an integer rotation number N to a noninteger N will actually lower the reliability.
In summary, a high reliability of spike timing requires a high derivative of the input command and a constant rotation number in the phase plane. This is obtained when a cell fires at a point in the phase plane where it is situated in a large area of an Arnold tongue, and at a minimum distance in the order of the standard deviation (SD) of the noise away from other Arnold tongues. In this way, the noise is unable to move it sufficiently in the phase plane to cause it to shift between rotation numbers. The ideal situation is therefore 1:1 phase locking.
Relationship between reliability and rotation number
One of the analytical predictions of the preceding section was
that there should be a connection between integer rotation numbers and
reliability. To examine this closer, we compiled the data of 124 sine
wave current stimulations performed on 7 different neurons and plotted
the reliability, P, of each stimulation as a function of the
rotation number, N (Fig. 6,
). Because of the pronounced low-pass characteristic of the neurons,
we were not able to obtain rotation numbers larger than two in the real
neurons.
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As predicted from the model, the reliability slowly increases until N = 1. At N = 1 the reliability can be high because of the large area in the phase plot, but the reliability is still dependent on the derivative of the stimulus. Both high and relatively low reliability can therefore be seen in Fig. 6 for N = 1. As N moves from N = 1 to N = 2, the reliability drops due to the small area of the Arnold tongue. At N = 2 high reliability can again be achieved.
When increasing the amplitude for frequencies above 1 Hz, the area between rotation number N = 1 and N = 2 is very small. This observation also fits perfectly with the nature of the Arnold tongues, as can be seen by making a horizontal cut through Figs. 4 and 5 (not shown). Around N = 1/2 (Fig. 6) there is a small increase in reliability. This local increase in reliability corresponds to stimulation frequencies above 15 Hz and can be explained as an increase of the Arnold tongue area for N = 1/2 at frequencies above 15 Hz (Fig. 4). For frequencies of stimulations above the values shown in Fig. 4, the tendency in the figure continues: the area of the N = 1/2 tongue increases as it broadens out further and the N = 1 tongue is moved to the right.
Overall, it seems that the predictions from the model about the relationship between the rotation number and reliability are confirmed by the experimental data.
Comparison of response of LIF model and real neurons
To evaluate the response of the LIF model more directly, we ran simulations using membrane parameters obtained in the intracellular recordings and compared the results of the simulations and the experimental values for reliability and rotation numbers. An example of this is shown for one cell in Fig. 7.
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An estimate of the membrane time constant, = 50 ms, was
obtained from the theoretical curve (Eq. 6) that was fitted
to the actual data (see Fig. 3C). The resting potential,
Vrest, was measured directly, whereas
the voltage threshold for spiking,
Vthres =
48 mV, was measured by
depolarizing the cell sufficiently for the cell to start firing.
Finally, we needed to determine the magnitude of the intrinsic noise,
C, in the real neurons. This required a comparison between
the numerical and experimental data. To do this we ran simulations of
the model with different magnitudes of noise, C, for fixed
frequency,
, but varied the sine wave amplitude, A, and
calculated the reliability for each set of parameters. These
simulations showed that the magnitude of noise in the model determined
the maximally achievable reliability. A high magnitude (SD > 1.5 mV) of noise made it impossible to obtain high reliability (P > 0.9). The simulations also allowed us to fit the
numerical data to the experimental and thereby get an estimate of the
magnitude of noise. Thus good fits were obtained using noise with SD of 0.9 mV. This seems to confirm the theoretical calculations of Manwani and Koch (1999)
, who estimated that the
intrinsic noise of the neurons should have a SD of 1.01 mV.
The result of the simulation (averaged over 50 runs) and the
experimental values for reliability and rotation numbers for a fixed
frequency ( = 3.125 Hz) but different amplitudes are shown in
Fig. 7. For fixed frequency, there was a linear relation between the
input current and the measured voltage amplitude
Vsin(
) = AA0R, so the experimental
results were scaled with a factor A0R. The solid lines show the
rotation number (bottom line; y-axis to right) and the
reliability (top line; y-axis to the left) for the
simulations, while the crosses shows the corresponding data from the
real experiment. There is a clear correspondence between the
simulations and the experiments. As for the real data, reliability (P) in the model increased with amplitude to a high value
(P > 0.6). This higher value was obtained and stayed
relatively stable when N = 1. P then
dropped, until it again increased when N moved though
noninteger numbers from N = 1 to N = 2. This profile is very similar to that seen in Fig. 6. When
N > 2, a comparison between simulations and real data
becomes impossible because of a lack of experimental data (see above,
Relationship between reliability and rotation number).
Similar data were obtained for other cells (n = 7).
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DISCUSSION |
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We have made a series of experiments stimulating spinal
interneurons with time-varying command currents and characterized the
reliability of spike timing (P). These interneurons are
located in a part of the lumbar spinal cord that is involved in
generating hind limb locomotion (Kiehn and Kjaerulff
1998; Kjaerulff and Kiehn 1996
). Our data
demonstrate that the reliability of spike timing in these interneurons
depends on the amplitude and/or frequency of the command input. In
general, P was low at low frequencies and amplitudes, and
high at high frequencies and amplitudes. These data were then compared
with simulations made using a LIF model with an additional noise term
added (Eq. 4). Using the LIF model and the quantitative
measure of the rotation number (N), we were able to explain
the changes in reliability observed when varying the amplitude and
frequency of the command current. Fitting the result of the LIF model
to the experimental data also enabled us to estimate the standard
deviation of the internal neuronal noise. We conclude that the model
captures many aspects of the reliability of spike timing in real
neurons. The simplicity of the model offers the possibility to use
analytical tools to understand the rather complex interactions between
input commands and evoked spike activity.
Comparison of theory and experimental data
The LIF model is extremely simple and does not provide an exact biophysical description of neuronal behavior. The LIF model, however, did reproduce the experimentally observed relationship between P and amplitude/frequency, suggesting that this complex relationship can be investigated in a simple neuronal model. With slight modification of the LIF model, it was able not only to represent the firing times but also to mimic the trajectory of the spikes and their afterpotentials, increasing its value for comparing with real recordings.
It has been suggested that, when stimulated with high frequencies and
amplitude, spike timing in the LIF model is too inaccurate to be used
in network models (Brown et al. 1999). This suggestion is based on comparisons with the more physiological Hodgkin-Huxley model. The problem of inaccuracies is not pronounced unless the neuron
was stimulated with a large proportion of excitatory postsynaptic potentials (EPSPs) with frequencies above 100 Hz, far outside the
frequency range used in the present experiments. As a first approximation, we therefore used a modified LIF model, which when stimulated with physiologically realistic input command captures the
basic firing behavior of the cell. This enabled us to analyze the
system using the concept of phase locking. Moreover, numerical simulations could be compared with real data to fit experimental parameters (input conductance, time constant, and internal noise).
Reliability of spike timing
The general finding from both the experiments and the LIF model is
that constant DC commands lead to low reliability of firing (P < 0.5), with a great amount of variability of
firing times for repeated stimuli (Fig. 1A). The
low-frequency sine wave command approaches the DC situation and
therefore also evokes low P values. However, very high
reliability (P > 0.95) could be obtained by increasing
the sine wave amplitude and frequency. These observations correspond to
what has been observed previously in two studies of cortical neurons
(Mainen and Sejnowski 1995; Nowak et al.
1997
). Mainen and Sejnowski (1995)
injected
command currents of Gaussian white noise with different values of
standard deviation SD and mean µ. They noticed an increase in
P from 0-0.4 to 0.8-0.9 as the SD of the injected signal
increased from 0 to 100 pA. This effect was not only mediated by the
increase in amplitude but also by concomitant increase in the
derivative of the input command (the derivative increases with larger
amplitudes). Nowak et al. (1997)
generated command
signals from natural synaptic activity recorded in vivo and injected
them into neurons in a slice preparation of the ferret visual cortex.
They found that P increased from 0.188 for a constant DC
current to 0.898 for the synaptic command currents containing
high-frequency components, indicating the possibility of high
reliability for naturally occurring synaptic inputs.
Despite the similarities in reliability between these different
studies, the precision of firing is different for cortical and spinal
neurons. Thus in cortical neurons the precision can be less than a
millisecond, while in spinal neonatal neurons it is in the order of
2-3 ms, because of the broad spikes in these latter neurons
(Fulton and Walton 1986). These differences apart, the
present study clearly shows that spinal interneurons are capable of
firing with high reliability when subjected to an appropriate stimulus command.
Phase locking
The general idea explored in this paper is the concept of phase locking of neuronal firing to an external command, in this case a current command. We have shown that there is a relationship between the rotation number (N) and the reliability (P). By examining phase locking, we can provide a mechanistic explanation for the relationship between the amplitude and frequency of the input command and the reliability of spike timing, which is not easily understood otherwise.
The analysis involves plotting the number of firings per period, the rotation number, as a function of the frequency and amplitude of the incoming signal, thereby generating areas of different rotation number. The large plateaus in the phase plot (Figs. 4 and 5), primarily the 1:1 phase locking, provide a robustness to the system that eliminates the influence of noise over a wide range of input commands.
Our findings show the importance of choosing the frequency components
and the amplitude of the input command in accordance with the time
constant and spike threshold of the neuron, to achieve high
reliability. It should be noted that the relevant command amplitude is
actually the relation between the resulting amplitude (RAA0) and the threshold for firing above
the resting membrane potential (Vrest Vthres), as can be seen by
nondimensionalizing Eq. 5 with
Vthres. A lower threshold for firing
would yield the same rotation number if the input command amplitude was
equally lowered.
Phase locking is a general phenomenon and can be found in many systems
experiencing a fluctuating input (see, e.g., Jensen 1998). These fluctuating inputs may be sine waves as in the
present case, but phase locking has also been observed for systems
driven by randomly fluctuating forces (Jensen 1998
).
The idea of thinking of neuronal firing in terms of phase locking is
not new (Keener et al. 1981; Knight
1972
). Recently Hunter et al. (1998)
applied
this approach to Aplysia motor neurons. They defined a ratio
called f/fDC, where
fDC is the mean firing rate resulting
from injecting a constant current, and f is the sine wave
frequency. By injecting sine wave currents of different frequencies
superimposed on the constant DC current used to determine fDC, they found that the highest
reliability was observed for frequencies equal to
fDC. Thus 1:1 phase locking between
the command frequency and the mean firing rate will optimize the
reliability of firing. Numerical calculations using a LIF model gave
the same results. These results are very similar to those obtained in
the present study. However, the type of phase locking is different in
the two studies. A constant suprathreshold depolarization in combination with a sine wave command will create areas in the Arnold
plot without phase locking, quasi-periodic states, which will even
further decrease the reliability outside of the integer rotation number
areas. For a more thorough discussion of this phenomenon, see
Alstrøm et al. (1988
, 1990
).
Reliability of spike timing during network activity
In our analysis of the reliability of spike timing, we restricted
the analysis to evaluating simple input commands: sine waves of
different amplitudes and frequencies. These input commands do not
directly resemble the synaptic inputs that a spinal interneuron experiences during normal behavior. They do capture, however, some of
the basic features of the synaptic network activity that converge onto
a rhythmically active interneuron during locomotion. Thus the membrane
potential of locomotor-related interneurons in the neonatal rat
oscillates between phases of relative hyperpolarization and relative
depolarization. These oscillations are slow with a frequency
corresponding to the step cycle (0.3-1 Hz) (Kiehn et al.
1996; MacLean et al. 1995
; Raastad et al.
1996
-1998
) and with average amplitude of the
oscillations in the range of 10-12 mV (Kiehn et al.
1996
; Raastad et al. 1996
). Distinct fast
synaptic potentials with faster frequencies are superimposed on the
slow oscillations. It is from the fast synaptic potentials that the spikes seem to be triggered in the depolarizing phase of the locomotor cycle.
Using the phase diagrams (Figs. 4 and 5), we can understand how the
different components of the locomotor-related membrane fluctuations
influence spike timing and can make predictions about the frequency
components of a presynaptic input that will be most reliably
transmitted. It is clear that a cell stimulated with sine-wave
frequencies around 1 Hz (the leftmost point in Fig. 5) and
lower occupies a very small area in the Arnold tongues. With the level
of internal noise (SD of 0.9 mV) determined in this and previous
studies (Manwani and Koch 1999), the neuron can be moved
between different Arnold tongues, thereby decreasing the reliability of
firing. Thus the slow membrane oscillations observed during locomotion
are bound to cause a very low reliability of spike timing. In contrast,
input commands with frequencies of >3 Hz will evoke a high reliability
of firing (rightmost point in Fig. 5). To obtain this, very large
positive voltage deflections are needed. Such large-amplitude voltage
deflections are not observed during natural synaptic activity in spinal
interneurons. However, if the neuron is depolarized so that the
membrane potential moves closer to spike threshold, the phase diagram
is shifted to the left along the x-axis, thereby reducing
the voltage fluctuations needed to give a high reliability. The slow
membrane voltage oscillations observed during locomotion can serve this
depolarizing function. Thus by combining the slow (around 1 Hz and
below) and the fast voltage (>3 Hz) oscillations is it possible to
obtain a high reliability of spike timing during locomotion. In this
context, it is worth noting that what we call fast oscillations are
relatively slow compared with the stimulus commands used in cortex
(Mainen and Sejnowski 1995
; Nowak et al.
1997
). Cortical neurons fire frequencies around 50-100 Hz,
while spontaneously active and locomotor-related interneurons fire with
frequencies around 1-10 Hz (Raastad and Kiehn 2000
;
Tresch and Kiehn 1999
, 2000a
). This
suggests that the frequency range for optimal reliability of firing is
precisely tuned to the natural firing range of the neuron.
So while the slow oscillations at first sight mainly seems to serve a
role as determining the difference between the preferred and
nonpreferred phase of firing during locomotion, they are as discussed
in the previous section a necessary condition for spinal interneurons
to fire with a high reliability. A high reliability of spike timing is,
however, only obtained if fast synaptic inputs are superimposed on the
slow oscillations as occur during locomotion. It is therefore likely
that, although single interneurons seem to fire very unreliably in
individual locomotor cycles (Raastad and Kiehn 2000;
Tresch and Kiehn 1999
, 2000a
), they might
in fact react with a high reliability of spike timing in response to
presynaptic activity. A high reliability of spike timing in individual
interneurons will tend to synchronize the activity across a group of
functionally related interneurons that are activated by a common
synaptic input (Usrey and Reid 1999
). Synchronized
reliable firing in a group of premotor interneurons will in turn lead
to a synaptic output, which may recruit motor neurons more efficiently
than an asynchronous synaptic activity. Moreover, model studies have
shown that synchronized inputs will increase the force output from
motor neurons more than asynchronous inputs for a given input rate
(Baker et al. 1999
). While the presence of synchronized
firing has not yet been investigated in spinal interneurons, there is
in fact indications that a synchronized output is "read out" to
motor neurons during locomotion and respiration (Christensen et
al. 1998
; Kiehn et al. 2000
; Kirkwood and
Sears 1978
; Tresch and Kiehn 2000b
). Altogether, this may suggest that a high reliability of firing in spinal
interneurons might serve an important role for the motor output
generated by the spinal locomotor network.
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ACKNOWLEDGMENTS |
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This work was supported by the Danish Medical Research Council, the Novo Nordisk Foundation, and NIH to O. Kiehn. P. Alstrøm acknowledges support from CORE.
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FOOTNOTES |
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Present address and address for reprint requests: O. Kiehn, Dept. of Neuroscience, Karolinska Institute, Retzius Vag 8, 17177 Stockholm, Sweden (E-mail: ole.kiehn{at}neuro.ki.se).
Received 21 February 2001; accepted in final form 2 July 2001.
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REFERENCES |
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