1Center for Neural Science and 2Courant Institute of Mathematical Sciences, New York University, New York, New York 10003; 3Department of Neurobiology, State University of New York, Stony Brook, New York 11794; and 4Mathematical Research Branch, National Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, Bethesda, Maryland 20814
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ABSTRACT |
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Smith, Gregory D., Charles L. Cox, S. Murray Sherman, and John Rinzel. Fourier Analysis of Sinusoidally Driven Thalamocortical Relay Neurons and a Minimal Integrate-and-Fire-or-Burst Model. J. Neurophysiol. 83: 588-610, 2000. We performed intracellular recordings of relay neurons from the lateral geniculate nucleus of a cat thalamic slice preparation. We measured responses during both tonic and burst firing modes to sinusoidal current injection and performed Fourier analysis on these responses. For comparison, we constructed a minimal "integrate-and-fire-or-burst" (IFB) neuron model that reproduces salient features of the relay cell responses. The IFB model is constrained to quantitatively fit our Fourier analysis of experimental relay neuron responses, including: the temporal tuning of the response in both tonic and burst modes, including a finding of low-pass and sometimes broadband behavior of tonic firing and band-pass characteristics during bursting, and the generally greater linearity of tonic compared with burst responses at low frequencies. In tonic mode, both experimental and theoretical responses display a frequency-dependent transition from massively superharmonic spiking to phase-locked superharmonic spiking near 3 Hz, followed by phase-locked subharmonic spiking at higher frequencies. Subharmonic and superharmonic burst responses also were observed experimentally. Characterizing the response properties of the "tuned" IFB model leads to insights regarding the observed stimulus dependence of burst versus tonic response mode in relay neurons. Furthermore the simplicity of the IFB model makes it a candidate for large scale network simulations of thalamic functioning.
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INTRODUCTION |
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Considerable attention has been focused in recent
years on the functioning of thalamic relays, because it has become
clear that the thalamus does not serve as a simple, machine-like relay of information to cortex (for recent reviews, see Sherman and Guillery 1996, 1998
). Instead the thalamus controls the extent and nature of information being relayed in a dynamic fashion that appears to be related to behavioral state and perhaps attentional demands. A good example is the lateral geniculate nucleus, the thalamic
relay of retinal information to visual cortex. Only 5-10% of synapses
on geniculate relay cells derive from retina. The rest derive from
nonretinal sources, including local GABAergic cells, feedback afferents
from visual cortex, and various pathways from the brain stem, and these
modulate the nature of retinogeniculate transmission (Sherman
and Guillery 1996
, 1998
).
In addition to the complexity of thalamic circuitry, the membrane
properties of relay cells contribute to the nature of the relayed
information. In particular, thalamic relay cells exhibit a voltage- and
time-dependent, low-threshold, transient Ca2+
conductance, that, when activated, allows Ca2+ to
enter the cell via T-type (for "transient")
Ca2+ channels, producing a transmembrane current,
IT, and leading to a large
depolarization known as the low-threshold Ca2+
spike. The inactivation state of IT
determines whether information is relayed to cortex in tonic
mode or burst mode (Jahnsen and Llinas
1984a,b
; Sherman 1996
). When the cell starts off
relatively depolarized (above roughly
60 mV for >50-100 ms),
IT is inactivated, and the relay cell
responds to an excitatory input [e.g., a retinal excitatory
postsynaptic potential (EPSP)] with sustained firing of unitary action
potentials. This is the tonic firing mode. However, if the cell is
hyperpolarized first (below roughly
65 mV for >50-100 ms), the
inactivation of IT is removed (i.e.,
IT becomes deinactivated), and now a
sufficient depolarization or EPSP will activate
IT. The result is a low-threshold
Ca2+ spike with a brief burst of 2-10 action
potentials riding its crest.
One of the keys to understanding how thalamic relays work is to
understand in more detail how the input/output properties of relay
cells are affected by the inactivation state of
IT. We sought to do this with both an
experimental and modeling approach. By recording from relay cells of
the lateral geniculate nucleus of the cat in vitro, we measured
input/output properties by injecting into the cell sinusoidal currents
that varied in amplitude, frequency, and mean level, and we performed
analogous input/output experiments on a minimal relay cell model to
test the degree to which the essential features of
IT accounted for relay cell responses.
In addition to providing an easily parameterized set of stimuli that lends itself to Fourier analysis, the use of sinusoidal current injection allows us to interpret our results in the context of the
spatial and temporal frequency analysis paradigm that has such a
successful history in visual systems neuroscience (for review, see
Shapley and Lennie 1985). Of course, our use of Fourier techniques is not based on an assumption of the linearity of relay neuron responses but simply reflects an historically preferred method
of extracting relevant measures of cellular response (see METHODS).
For the theoretical component of this study, we developed a minimal
"integrate-and-fire-or-burst" (IFB) neuron model. This model is
constructed by adding a slow variable representing the deinactivation
level of IT to a classical
integrate-and-fire neuron model (Knight 1972). The IFB
model is designed specifically to be as simple as possible while still
quantitatively reproducing much of the empirically observed properties
of the relay cells. One motivation for developing such a minimal model
is to simplify the parameter selection process. Furthermore, because
the IFB model is minimal, a detailed characterization of its response properties leads to insight regarding the stimulus dependence of burst
versus tonic response modes in thalamic relay cells. A final motivation
for development of the IFB model is to have a realistically tuned yet
computationally undemanding relay cell model that can be used in large
scale network simulations of thalamic function.
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METHODS |
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Experimental methods
We performed intracellular recordings with the whole cell configuration on thalamic relay cells of young cats (5-8 wk of age) in compliance with approved animal protocols. We used a thalamic brain-slice preparation containing the lateral geniculate nucleus. Briefly, the animals were anesthetized deeply with 25 mg/kg ketamine and 1 mg/kg xylazine and a block of tissue containing the thalamic region was removed and placed in cold, oxygenated slicing solution containing (in mM) 2.5 KCl, 1.25 NaH2PO4, 10.0 MgCl2, 0.5 CaCl2, 26.0 NaHCO3, 11.0 glucose, and 234.0 sucrose. Thalamic slices (250-300 µm) were cut in a coronal or sagittal plane with a vibrating tissue slicer and placed in a holding chamber (30°C) for >2 h before recording. Individual slices were transferred to a submersion-type recording chamber maintained at 30°C and continuously perfused with oxygenated physiological solution containing (in mM) 126.0 NaCl, 2.5 KCl, 1.25 NaH2PO4, 2.0 MgCl2, 2.0 CaCl2, 26.0 NaHCO3, and 10.0 glucose, all at pH 7.4.
We used an Axoclamp 2A amplifier to obtain current-clamp recordings from geniculate relay neurons in the A-laminae, and we continuously monitored the bridge balance throughout the recordings. The recording pipette solution contained (in mM) 117.0 K-gluconate, 13.0 KCl, 1.0 MgCl2, 0.07 CaCl2, 0.1 EGTA, 10.0 HEPES, and 0.5% biocytin. Data were digitized, stored on-line using Axotape software (Axon Instruments), and also recorded onto VHS tape for off-line analysis. Current injection through the recording electrode consisted of a sinusoidal waveform with an AC component (I1) that varied in both amplitude (50-800 pA) and frequency (0.1-100 Hz). The DC component (I0) of the current waveform was altered to manipulate the firing mode of the neuron (i.e., burst vs. tonic). All experimental records of membrane potential of relay neuron in whole cell mode have been adjusted to account for a 10-mV junction potential.
Fourier analysis of experimental and theoretical responses
Customized user M-files were written for MATLAB 5.2.0 (The
MathWorks) to perform data analysis using an SGI Challenge
supercomputer that runs the IRIX operating system. For each stimulus
condition, a periodic histogram (qk,
k an integer, 0 < k < N
1, n = 64 bins) was constructed that tallied over
c cycles of period T the number of action
potentials (qk) evoked by the experimental
or model relay neuron at each of N blocks of phase relative
to the applied current's period. Accounting for the number of cycles
recorded and the time represented by one bin of phase (i.e., for 64 bins, 1 bin is
/32 rad), we generated the (periodic) poststimulus
response histogram (PSTH) defined by Qk
qkN/cT. A discrete Fourier
transform of this PSTH was performed, leading to a set of N
complex valued numbers,
n,
given by (Press et al. 1992
)
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For clarity of figure presentation, we also normalized the raw
histogram, qk, by the total number
of spikes during the trial (qtot). The
result is a spike phase density histogram (SPDH), defined by
k
qk/qtot.
This SPDH has unit area and represents the likelihood of observing an
action potential at a particular phase of the applied current.
IFB model
GENERAL FEATURES.
The IFB model is constructed by adding a slow variable to a classical
integrate-and-fire model neuron. The slow variable, h,
represents the inactivation of the low-threshold
Ca2+ conductance, which involves T-type
Ca2+ channels and produces a transmembrane
current, IT. The model equations are
(Rinzel 1980)
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PARAMETER SELECTION.
Standard parameters were selected for the IFB model in the following
fashion. First, experimental observations indicated that the resting
membrane potential of the relay neurons recorded in vitro was 75 to
65 mV. In the absence of applied current
(Iapp), the leakage term
(IL) exclusively sets the resting
potential of the neuron model. We thus set
VL to
65 mV. A second experimental observation is that the relay neurons recorded in vitro are
hyperpolarized sufficiently at rest so that
IT is deinactivated. Indeed, quiescent relay neurons in the slice responded to a brief depolarization with a
burst of action potentials. For this reason, we set
Vh to
60 mV, ensuring that the threshold
for activation (and deinactivation) of
IT is several millivolts greater than
VL, as observed experimentally. This
value for Vh also roughly corresponds to
the observed threshold for activation of bursts.
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RESULTS |
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Current-clamp recordings from a total of 12 relay neurons from the
lateral geniculate nucleus of the cat were included in the present
study. All cells displayed physiological characteristics consistent
with healthy relay neurons, including a hyperpolarization-activated sag
current (Ih), burst discharge, and
overshooting action potentials. Cells included in this study had an
average resting membrane resting potential of 67.8 ± 4.1 mV
(mean ± SD) and an input resistance that averaged 153.0 ± 38.0 M
.
The preferred firing mode of the relay celltonic or burst
was
controlled largely by constant current injection from the recording pipette. This constant current injection is referred to in the following text as I0, which we varied
between experiments from
400 to 800 pA. On top of this constant
current, we injected various other currents, usually sinusoidal at
various frequencies and amplitudes. At relatively depolarized membrane
potentials (i.e., more positive values of
I0), the cell fired to the sinusoidal current in tonic mode, whereas at relatively hyperpolarized membrane potentials (i.e., more negative values of
I0), the cell fired in burst mode; at
intermediate levels of membrane potential, responses often consisted of
a burst followed by tonic firing. It is noteworthy that in all cases
when we saw both response modes to a cycle of the current injection,
burst firing always preceded tonic firing.
General responses to current injection
In Fig. 1A, the results
of current steps injected into the relay cell experimentally are shown.
When the membrane potential (Vm) was
adjusted initially to 58 mV (Vhold;
I0 = 230 pA), the neuron discharged in
tonic mode in response to a short current pulse (200 pA). However, at a
more hyperpolarized Vhold (
77 mV), a
depolarizing current step (50 pA) evoked a transient burst of high-frequency action potentials. The IFB model produces both tonic and
burst responses (Fig. 1B) that are similar to those produced
experimentally. Figure 1B also shows the inactivation gating
variable of the simulated low-threshold Ca2+
current (denoted by h; see METHODS) dropping
from unity to near zero shortly after the onset of the current pulse.
In the IFB model, the time scale of this inactivation determines the
length of the burst event. The value of h remains near zero
(representing inactivation of the low-threshold
Ca2+ conductance) until the depolarizing pulse
ends, at which point the membrane potential drops below
Vh, the threshold for deinactivation of IT, and h recovers
toward 1.
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Figure 2A presents
experimental recordings that illustrate firing patterns of a geniculate
relay neuron during sinusoidal current injection over a range of
stimulation amplitudes (I1) while the
modulation frequency is held constant at 1 Hz. In Fig. 2A,
left, the neuron is in tonic mode, because of a more depolarized I0 of 335 to 498 pA, whereas in the
right column, the cell is hyperpolarized because
of a more hyperpolarized I0 of 4 to
136 pA and responds with burst discharges. When the neuron responds in tonic mode, the average number of spikes/cycle increases as I1 is increased. In contrast, in burst
mode, the neuron responds with 1 burst/cycle over a wide range of
I1, and the number of spikes/burst
remains relatively constant at 7 or 8. An exception to this occurs at
the lowest-modulation amplitudes, where the neuron bursts once for
every several cycles of applied current (Fig. 2A, top
right). In this paper, we will refer to such behavior as a
subharmonic burst response. In addition to these
representative patterns of cell responses to 1 Hz stimulation, we also
observed superharmonic burst responses (2 bursts/cycle),
subharmonic tonic responses, and burst followed by tonic responses.
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Figure 2B presents a series of IFB model calculations for comparison with Fig. 2A. These calculations were performed using identical cellular parameters (see Table 1), whereas applied current parameters were chosen in qualitative agreement with the experimental conditions used in Fig. 2A. The simulations reproduce many salient features of the experimental recordings. For example, the IFB model responds in tonic mode to more depolarizing mean applied current (I0) and in burst mode to more hyperpolarizing I0. In tonic mode, the average number of spikes/cycle increases as a function of I1, whereas in burst mode, 1 burst/cycle is observed over a wide range of I1. In addition, the IFB model produces 7-8 spikes/burst, relatively independent of I1. Although the IFB model reproduces the overall pattern of responses to a range of stimulus conditions in both tonic and burst responses, it does not reproduce the subharmonic responses observed at low modulation amplitude (cf. Fig. 2, A and B, top right; see DISCUSSION).
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Figure 3A consists of recordings from the same neuron as Fig. 2A, but here sinusoidal current injection over a range of I0 values and stimulation frequencies is applied while I1 is fixed. When the holding Vm is adjusted so that the neuron is in a tonic firing mode (depolarized I0; left), the cell exhibits tonic firing in response to I1 frequencies of 0.3, 1, and 3 Hz, and is unresponsive to 10 Hz. The average number of spikes/cycle decreases in tonic mode as frequency is increased. However, when the neuron is in burst mode (hyperpolarized I0; right), 1 burst/cycle (5-8 spikes/burst) is observed in response to 0.3 and 1 Hz until subharmonic responses are evoked at 3 Hz; no response was seen at 10 Hz. The subharmonic responses were observed most commonly in response to a frequency of 3 Hz, occasionally at 1 Hz. Many other cells did respond to 10 Hz stimulation, especially in tonic mode (see following text). In 4 of 12 cells, superharmonic responses of 2 bursts/cycle were observed at low stimulation frequencies (0.1-0.3 Hz; see following text).
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Figure 3B presents a series of IFB model calculations that reproduce many salient features of the experimental results shown in Fig. 3A. For example, when the model responds in tonic mode (left), the average number of spikes per cycle decreases as frequency increases. Yet when the model responds in burst mode, 1 burst/cycle is observed over a wide range of frequencies of I1. At the cutoff frequency of 10 Hz, both the relay neuron and the IFB model are unresponsive, and the effect of the mean applied current, I0, on the mean membrane potential can be seen clearly (see Fig. 3, bottom). The IFB model thus qualitatively reproduces the neuron responses over a range of stimulus frequencies with the exception of subharmonic responses.
Superharmonic burst responses
As mentioned in the preceding text, burst responses observed at low frequency were primarily 1 burst/cycle (1:1). However, superharmonic burst responses were sometimes observed at low frequencies. Examples recorded from two geniculate relay cells of single (1:1) and double (2:1) burst responses at low frequency are shown in Fig. 4, A and B, respectively. Figure 4, left, shows responses to a more hyperpolarized I0, and the right shows responses to a more depolarized I0 (see legend for details). The result is pure burst responses on the left and burst followed by tonic responses on the right. Because superharmonic burst responses were seen in response to the lowest temporal frequencies tested, which was 0.1 Hz, the prevalence of these two response types in our data have been quantified in the following way. There were a total of 56 trials collected from 12 different cells that exhibited bursts at 1-3 Hz. Of these, we quantified the response type at the lowest frequency tested, 0.1 Hz. The majority (61%) were 1:1 (i.e., 1 burst/cycle as in Fig. 4A). Superharmonic bursts (i.e., 2:1 as in Fig. 4B) were observed in 14% of the trials, whereas 3 bursts/cycle (3:1) were never observed. Interestingly, in the remaining 25% of trials, no response at all was observed at 0.1 Hz. However, at a higher frequency (0.3 Hz), only 3 of 56 trials (5%) exhibited no response, indicating that burst mode for some neurons has a genuine band-pass character to varying temporal frequency (see following text).
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Responses as a function of frequency and amplitude of current injection
The filtering characteristics of relay neurons differ depending on
the firing mode of the neuron. Figure 5
summarizes responses of two relay cells in both burst (hyperpolarized
I0) and tonic mode (depolarized
I0) to different levels of
I1 and frequency of current injection.
The responses have been categorized into the following four classes:
burst followed by tonic (filled circles), only burst (open circles,
left) or only tonic (open circles, right), subharmonic burst (asterisks, left) or subharmonic tonic
response (asterisks, right), and no response (dashes).
Figure 5A, left, summarizes the responses of one of these
neurons in burst mode. At low I1 (50 pA), this neuron responded to 3 Hz with subharmonic bursts (asterisks)
and gave no responses to higher or lower frequencies. With increasing
I1 (100 pA), the cell now responds 1:1
at 0.3 and 1 Hz, subharmonic at 3 Hz, but is still unresponsive to 0.1 Hz. With further increases in I1 (200 pA), the neuron now responds to the full range of 0.1-3 Hz, but is
unresponsive to 10 Hz. Thus burst responses of this cell show band-pass
filtering for lower I1 and low-pass
filtering for higher I1in the latter
case, the high-frequency cutoff increases to 10 Hz for
I1 = 300pA. Similar results from a
second neuron responding in burst mode are presented in the Fig.
5B, left.
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In tonic mode (Fig. 5, right), similar values of I1 and frequency produce a different pattern of responses than observed in burst mode. First, all responses in tonic mode exhibit either low-pass or broadband filtering; there is no band-pass filtering because the cells always respond well to the lowest frequencies tested. Second, the high-frequency cutoff was greater in tonic than in burst mode, and for high values of I1, there may be no observable cutoff frequency (Fig. 5A, top right). In this example, no high-frequency cutoff is observed because I0 was greater than the rheobase of neuron, whereas in the other example (Fig. 5B, right), I0 was less than the rheobase. It is also notable that subharmonics in tonic mode, when apparent, generally occurred before the cutoff frequency.
Fourier analysis of relay cell and IFB model responses
To compare quantitatively the different consequences of burst and
tonic firing modes on relay cell responses to sinusoidal current
injection, we performed Fourier analysis of the intracellular recordings (see METHODS). SPDHs were constructed from
experimental recordings, and Fig. 6 shows
examples of how this is done for several different stimulus conditions
producing burst, tonic, or burst followed by tonic firing (see
following text for details of stimulation parameters). Figure
6A shows responses to four cycles of the injected current.
Figure 6B shows these responses aligned on a cycle-by-cycle
basis, and Fig. 6C shows SPDHs constructed by assigning each
spike to 1 of 64 bins depending on the value of its phase with respect
to I1. In the resulting histograms, the spike density, , approximates the likelihood of a neuron firing
an action potential at a given phase,
, of
Iapp. To quantify the dependence of the SPDHs on
stimulus parameters (f, I0,
and I1), discrete Fourier transforms
of such histograms were performed leading to the assignment of four
response measures: F0, the mean firing
rate; F1, the stimulus- or
modulation-driven component of the response;
P1, the phase advance or lag of the
modulation-driven response; and
, the nonlinearity of the response.
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Figure 6C shows examples of SPDHs that are representative of
our results from relay cells exhibiting tonic, burst, and burst followed by tonic responses obtained at 0.3 Hz. When stimulus parameters were such that the neuron responded in tonic mode
(I0 = 410 pA and
I1 = 150 pA), the SPDH approximates
the shape of a rectified cosine. When stimulus parameters were such
that the neuron responded in burst mode
(I0 = 4 pA and
I1 = 50 pA), the SPDH does not
approximate a rectified cosine but rather shows a sharp peak near
=
0.25 (i.e., a 90° phase advance). These features are
combined in a SPDH generated from intracellular records that exhibited
burst followed by tonic responses (Fig. 6C). The burst and
tonic portions of this response are separated by ~10°, and because
the burst always proceeds the tonic response of each cycle,
there is a gap in the SPDH.
INTERPRETATION OF RESPONSE MEASURES.
Because many of the results to follow involve the relationship between
the response measures, F0,
F1,
P1, and , it is instructive to
review our expectations of these measures. Imagine an idealized tonic
response of a neuron to be proportional to the rectified cosine (see
Fig. 6C), most commonly
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FOURIER ANALYSIS OF EXPERIMENTAL DATA. With this background, Figs. 8 and 9 present the results of Fourier analysis of the responses of a population of relay cells showing burst and tonic responses to sinusoidal current injection. Only data involving pure burst or tonic responses (i.e., no burst followed by tonic responses) are shown here, which means that these data reflect mostly small values of I1 and/or extremely hyperpolarized or depolarized values of I0. Figure 8, A-D, summarizes experimental results from 84 trials (8 different cells) in which only burst responses were observed. When plotted in units of spikes/second, the mean firing rate (F0) is approximately half the value of the modulated response (F1), and both are band-pass with peaks at 3 Hz (Fig. 8A). These data subsequently are plotted in units of spikes/cycle (Fig. 8B), showing that, at lower frequencies, a relatively constant number of spikes/cycle are observed, and in the majority of cases, these result from one burst/cycle.
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FOURIER ANALYSIS OF THE IFB MODEL.
Figures 10 and
11 present the results of Fourier
analysis applied to IFB model responses. As in Figs. 8 and 9,
I0 and
I1 are chosen to ensure pure tonic or
burst responses from the model (burst followed by tonic responses are
considered in the following text). For burst firing, there is a
remarkable, quantitative equivalence between experiment (Fig. 8) and
theory (Fig. 10). One subtle difference is that the IFB model phase
locks more precisely than actual relay cells (cf. 3-Hz SPDHs in Figs.
8E and 10E). This may be due to the presence of
noise in the voltage recordings that is absent in the IFB model.
Indeed, when the simulated applied current is supplemented with
Gaussian noise (mean amplitude of 0 pA and a variance similar to that
of I1), phase locking by the IFB model is strongly attenuated (data not illustrated), and the nonlinearity index () is reduced. IFB model tonic responses (Fig. 11) are also comparable with experimental records (Fig. 9), though a notable exception is a gradual drop in
at 30-100 Hz that is seen
experimentally but not reproduced (cf. Figs. 9D and
11D). Again, this difference may reflect the presence of
noise in the experimental recordings because we would expect a uniform
amount of spike jitter (in units of time) to be more apparent in SPDHs
obtained during high-frequency stimulation (when bins of phase
correspond to shorter time intervals). Because in the in vivo
experimental condition there is more synaptic (and other sources of)
noise than encountered in vitro, we would expect less phase locking and
more linearity in vivo than we have seen experimentally here
(Carandini et al. 1996
). Nonetheless, phase-locked relay
neuron responses to drifting sinusoidal contrast gratings have been
observed in vivo (Reich et al. 1997
, 1998
).
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Phase plane portrait of the IFB model and high-frequency roll off in burst mode
As the cutoff frequency is approached, burst responses of relay cells gradually decline or roll off (e.g., Fig. 8, A and B). One obvious reason for this is that some neurons exhibit subharmonic bursts in this frequency range, and this would cause a decrease in spikes/second and spikes/cycle. In a subset of trials that exhibited bursting in the range of 1-3 Hz, we found subharmonic (1:N) bursting and 1:1 bursting to be nearly equally prevalent at 3 Hz (36 and 43%, respectively), suggesting for some cells another reason for this roll off, one that is predicted by our IFB model. Figure 12A shows burst responses from the IFB model that demonstrate the roll off in F0 during 1:1 bursting. At 2 Hz, the model responds with 1 burst/cycle, and each burst is composed of 6 spikes/burst. However, at 6 Hz, where 1 burst/cycle also is seen, only 2 spikes/burst are evoked. Also note that the maximum deinactivation levels (hmax) of IT achieved during the hyperpolarizing phase of the applied current is much greater at 2 than 6 Hz (Fig. 12B). The greater hmax at 2 Hz thus leads to a larger evoked low-threshold Ca2+ spike, which, in turn, evokes more action potentials.
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Figure 12C presents phase-plane portraits of the IFB model
at 2 and 6 Hz. This helps to clarify the relationship between membrane potential (V) and thus spike discharge, and the inactivation
gating variable, h, at both frequencies. The solid line and
arrows show the trajectory in the (V, h) plane that is
repeated from cycle to cycle. The threshold for
IT
(Vh, dashed line),
Vreset (dot-dashed line), and
V (dotted line) also are indicated.
During the hyperpolarizing phase of the applied current, V
eventually drops below Vh, causing
h to increase (arrow 1). Eventually the current reverses,
leading to depolarization; however, because V is still less
than Vh, h continues to
increase (arrow 2). When the membrane potential crosses
Vh,
IT activates, h begins to
drop, and IT depolarizes the model
neuron until the spike threshold,
V
, is reached (arrow 3). A series
of action potentials are evoked (arrow 4) and h decreases
until the sum of IT and the applied
current are no longer large enough to bring the membrane potential
above threshold. When the applied current again reverses, the membrane
potential hyperpolarizes, V eventually drops below
Vh, and the periodic burst response repeats.
Figure 12D shows a plot of the frequency-dependence of
hmax. Because the time constant for
inactivation of IT is smaller than the
time constant for its deinactivation (see METHODS),
hmax decreases as frequency increases.
This, in turn, means that the size of the evoked low-threshold
Ca2+ spike and the number of action potentials
riding its crest will decrease at higher frequencies. It is thus this
decline in hmax as a function of
frequency that leads to the high-frequency roll off in the IFB model
burst response. Although we do not have direct access the gating
variable, h, in our experimental recordings, the open
squares and diamonds in Fig. 12E show the number of
spikes/burst exhibited by two relay neurons that burst 1:1 at all
frequencies tested. This qualitatively matches roll off in spikes/burst
exhibited by the IFB model (open circles) using standard parameters,
although a better fit for these particular trials was obtained by
increasing h+ from 100 to 300 ms, which
resulted in a lower cutoff frequency (filled circles).
Dependence of response measures on modulation amplitude
In the analyses summarized in Figs. 8-11, I1 was fixed and small enough to avoid burst followed by tonic responses, and I0 controlled the response mode by being either relatively depolarizing (for tonic firing) or hyperpolarizing (for burst firing). However, in both modes, the quantitative value of response measures depends on I0 and I1. This is true for both actual relay cells as well as the IFB model. Figure 13, A-C, left, presents the frequency dependence of the fundamental response F1 for 11 relay neurons using either low I1 (50-200 pA; filled squares) or high I1 (300-500 pA; open triangles), and the right panels show the comparable responses from the IFB model. These results are pooled according to the firing mode based on I0 so that Fig. 13A (I0 high) presents responses that are predominantly tonic, whereas Fig. 13C (I0 low) presents predominantly burst responses, and Fig. 13B (I0 medium) includes many burst followed by tonic responses.
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Note that the fundamental response is generally greater in Fig. 13 for the higher I1 than for the lower regardless of response mode. During tonic firing (Fig. 13A), there is little if any frequency dependence in the response augmentation with higher I1. However, when bursting occurs (Fig. 13, B and C) the degree of augmentation for large I1 can be frequency dependent. For example, the experimental responses show a greater F1 for the larger I1 at lower frequencies, but there is little difference in F1 at 10-100 Hz (note the overlapping error bars in Fig. 13B, left, for these frequencies). Qualitatively, this pattern is matched in the IFB model (Fig. 13B, right), which shows a clear peak in the augmentation of F1 with I1 at 3 Hz and no difference in response to higher frequencies. In experimental observations of burst mode (Fig. 13C, left), there is a clear distinction between low and high I1.
Because with the fairly stable number of spikes/burst we observe, the response in spikes/seconds becomes increasingly small at lower frequencies. The IFB model qualitatively exhibits these same features (Fig. 13C, right), and the response function for F1 reproduces this band-pass characteristic. However, for the most part this attenuation is "artificial," that is, it simply reflects the fact that the response is measured in spikes/second. To clarify this point, the solid squares in Fig. 13D redisplay in units of spikes/cycle F1 for the low I1 case shown previously (Fig. 13C, solid squares). In these units the neuronal response is low-pass rather than band-pass (cf. Fig. 8, A and B).
At the lowest values of I1 tested, we found that 6 of 11 neurons were unresponsive to the lowest frequencies tested but responsive to higher ones (see Fig. 5), while the remaining 5 responded to even the lowest frequencies. The open circles in Fig. 13D show the average F1 in units of spike/cycle. The F1 is attenuated at 0.1 Hz relative to the 0.3- and 3-Hz response, even when measured in units of spike/cycle (Fig. 13D, open circles). This occurs because approximately half of the trials were genuinely band-pass, that is, for this low value of I1, 50% of the neurons did not respond at 0.1 Hz even though they did respond with 1 burst/cycle at 0.3 Hz.
Parameter studies of IFB model responses
To give a sense of the dynamics of transitions between tonic and burst modes of firing as parameters vary, Fig. 14, A-C, presents a series of raster plots calculated using the IFB model in which I0 is varied while the frequency and cellular parameters are fixed, and amplitude of modulation increases from A to C (see legend of Fig. 14 for details). In Fig. 14A, where I1 is low, as I0 becomes more depolarizing, one sees a transition from silence to burst responses to another silent zone and then to tonic responses. Another transition in Fig. 14A is observed near I0 = 1.4 µA/cm2, where tonic spikes occur throughout the stimulus cycle (as opposed to the interrupted spiking). This happens when I0 is sufficiently depolarizing to ensure that, when added to I1, the model cell's membrane potential is always continuously driven above threshold for action potentials. The open circles in Fig. 14A show the phase of the centroid of the response (P1) plotted relative to the time of an applied current maximum (vertical dot-dashed line). Burst responses are phase lagged when first recruited at low I0 but then advance in phase as I0 is increased. Tonic responses evoked as I0 becomes more depolarizing consistently have a phase near zero.
|
By allowing a close look at spike timing as I0 is varied, the raster plot of Fig. 14A complements the Fourier analysis presented previously. For example, the phase of individual tonic spikes generally advances with increasing I0. The new spikes that occasionally are recruited as I0 increases generally appear in the late phase of the stimulus cycle, that is, when the applied current is decreasing. During interrupted tonic spiking, there is no consistent increase in P1 as a function of I0 because the gradual phase advance of individual spikes is balanced by new spikes arriving later in the cycle.
The gap between burst and tonic responses in Fig. 14A
indicates a range of I0 for which the
IFB model does not respond. Such a gap also was observed experimentally
(data not shown) and is present in the IFB model when
I1 is small enough so that, over a
certain range of I0, the membrane
potential oscillates in response to the applied current without
crossing either of the two thresholds, V or
Vh. This gap is reduced or
nonexistent with larger values of I1
(Fig. 14, B and C). Figure 14B shows
the result of larger I1, and here
burst responses appear more robust, beginning at lower values of
I0 and extending into higher values.
Comparing Fig. 14A and 14B, one also sees that
increased I1 extends the range of
interrupted tonic firing to higher levels of
I0. In Fig. 14C, I1 is made even larger, leading to
burst followed by tonic responses over a large range of
I0. The transition point between
interrupted and continuous tonic spiking has moved upward beyond
I0 = 3.0 µA/cm2 and is no longer visible.
Figure 14, D-F, shows rastergrams of the IFB model burst
responses as a function of the frequency of the applied current where the abscissa indicates the phase () of each spike. Because of the
compression of time in this coordinate at lower frequencies, the burst
events appear to narrow, even though their duration is relatively
constant. Figure 14, D-F, varies only in
I0, which is increasingly depolarizing
from D to F. Figure 14D thus
illustrates only burst responses, and comparing the open circles and
vertical dot-dashed line here reveals that the phase advance of these
responses gradually changes to a phase delay as the frequency increases from 0.1 to 10 Hz (cf. Fig. 10C).
In Fig. 14, E is similar to D, but
I0 is depolarizing, and interrupted
tonic responses are evoked. Between 0.1 and 1 Hz, the fraction of the
stimulus cycle occupied by tonic spikes is approximately constant. The
IFB model here is responding relatively linearly (e.g., = 0.15 at 0.3 Hz; see Fig. 11D). From 0.1 to 6 Hz,
P1 is approximately zero (cf. Fig.
11C) even though the phase of the first tonic spike advances
considerably in the range of 1-6 Hz. Above 6 Hz, the model responds
with 1 spike/cycle, and P1 follows the
phase lag of this solitary spike until the cutoff frequency is reached.
Here the response in nonlinear (e.g.,
= 0.94 at 10 Hz; see
Fig. 11D). The low-pass character (no response at
30 Hz)
of these tonic responses reflects the fact that
I0 is subthreshold for action
potentials here. For comparison, Fig. 14F presents a calculation identical to E except that
I0 is increased to a level that is
superthreshold. In this case, the IFB model responds with tonic spikes
even at 100 Hz. Strong phase locking is seen at higher frequencies with
low S:C ratios that change discontinuously with frequency.
The dual threshold character of the IFB model and the influence of
I0 on the functional dependence of
F0 and
F1 on
I1 is illustrated further in Fig.
15. With stimulation frequency fixed at
2 Hz, I0 is more hyperpolarizing in
the successive rows of Fig. 15, so that A exhibits purely
tonic responses, E exhibits only burst responses, and
B-D show gradual shift from pure tonic, to burst followed by tonic to pure burst responses. Voltage trajectories are presented for two representative values of I1,
the column labeled I1 Low presenting simulations with a modulation amplitude that is small in
comparison with the I1 High
column. These voltage trajectories are superimposed on dotted and
dashed lines, indicating the tonic spike threshold
(V) and
IT threshold
(Vh), respectively.
|
In Fig. 15, the arrows labeled Vss
indicate the membrane potential that would be approached (in the
absence of spiking) on the basis of the DC bias current alone so that
Vss = VL + I0/gL, where VL is resting membrane potential
and gL is leakage conductance of the
model. Indeed, the relationship among
Vss,
V, and Vh determines the functional form of
F0 and
F1 as a function of I1. For example, in Fig.
15A, I0 is depolarizing
enough that the steady-state voltage,
Vss, is greater than the threshold for
action potentials (Vss > V
). Thus low
I1 gives continuous tonic spiking, whereas
high I1 gives interrupted tonic spiking.
F0 is thus nearly constant with
I1, whereas
F1 increases as a function of
I1. Compare this with Fig.
15B, where I0 is less
depolarizing, so that the steady-state voltage is near the action
potential threshold (Vss = V
). In this case, low
I1 gives low-frequency interrupted tonic spiking,
whereas high I1 results in a burst followed
by tonic response, because the minimum membrane potential during a
cycle is now less than Vh, thereby
resulting in enough IT deinactivation
to allow the ensuing depolarizing half-cycle of the stimulus to evoke
an early burst. Here F1 is once again a nearly linear function of I1;
however, F0 is no longer relatively constant as in Fig. 15A but rather has a similar slope as
F1. The voltage trajectory for the
low I1 case in Fig. 15C shows a
null response followed after a threshold
I1 is exceeded by a fairly linear
growth in both F0 and
F1 with increasing
I1. Here V oscillates between V
and
Vh for low values of
I1 (see Fig. 15C, middle) and, consequently, the IFB model does not respond for this particular value of I0, resulting in a gap
between burst and tonic responses visualized previously in the IFB
model raster plot (Fig. 14A). In Fig. 15D,
I0 is such that that the steady-state voltage is
near the burst threshold (Vss = Vh, and a consequence of this is
that even the lowest nonzero value of
I1 elicits a burst response. For Fig.
15E, the response is entirely composed of bursts, and F0 and
F1 respond similarly with increasing
I1, showing a fairly sharp threshold
when I1 is sufficiently large to
recruit bursts followed by a slight increase in firing.
The dependence of response measures on the value of both the mean
and modulated applied current is summarized in Fig.
16, which presents Fourier analysis of
IFB model responses to 0.3-Hz stimulation over a range of
I0 and
I1 values. The front edge of the
surface plot of F0 in Fig.
16A, where I1 is zero, is
the current-frequency relation for the IFB model in tonic mode, that
is, the classical leaky integrate-and-fire current-frequency relation
(see APPENDIX). The rheobase occurs near 1.1 µA/cm2, and, because we have not included an
absolute refractory period in the spike generating mechanism of the IFB
model, the current-frequency relation does not saturate and is
approximately linear (except near the onset of repetitive firing),
reaching 80 spikes/s at 4 µA/cm2. When
I0 is hyperpolarizing, there is a
value of I1 above which the IFB model
can respond. In the projection of the surface onto the horizontal
plane, the leftmost region of nonzero response runs from near
I0 = 0 µA/cm2
and I1 = 0 µA/cm2) to respective values of 4 and 3, reflecting the tradeoff between I0 and
I1 that allows V, during
the depolarizing phase of the applied current, to exceed
Vh (see APPENDIX). These
responses are due primarily to bursts, a fact reflected in the plots of
P1 and
(Fig. 16, C and
D), which show that the response near this interface is
nonlinear and phase advanced. Figure 16B plots the
dependence of F1 on
I0 and
I1, which follows the same general
trends as indicated in Fig. 16A, the one major exception
being that F1 is much less than
F0 when
I0 is high and
I1 is low (bottom right of
graph). The largest discrepancies between
F0 and
F1 occur when the IFB model is tonic
spiking in a modulated, but continuous, manner. In Fig. 16D,
the peaks of elevated
that occur when
I0 > 1.1 µA/cm2 and reflects phase locking that occurs
when the firing frequency of the IFB model (determined solely by the DC
current, I0) are such that an integral
number of spikes occur in one stimulus cycle. This phase locking is
artifactual in the sense that when I1 = 0 the stimulation frequency of 0.3 Hz is relevant only because it
defines the Fourier fundamental frequency.
|
Figure 17 is identical to Fig. 16
except that the stimulus frequency is 3 Hz. One distinction between the
low- and high-frequency results can be seen by comparing the surface
plots for P1 (Figs. 16C vs.
17C). In the 0.3-Hz case, the responses are always phase advanced. However, in the 3-Hz case, both phase advance and phase lag
are observed depending on I0 and
I1 (as in Fig. 14A). There is also a significant region of the response surface in which at 3 Hz is elevated compared with 0.3 Hz (Figs. 16D vs.
17D). This reflects phase locking that is more prominent at
the higher frequency.
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DISCUSSION |
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Our results provide further support for the functional
differentiation between burst and tonic response modes in thalamic relay neurons. From intracellular recordings of relay neurons in the
cat's lateral geniculate nucleus in a thalamic slice preparation, we
have used Fourier analysis to quantify the response properties of relay
neurons in different discharge modes. We concentrated on the temporal
tuning of four relay cell response measures:
F0, F1,
P1, and . We found that the
temporal tuning of relay neurons in cat is highly dependent on response
mode (i.e., burst vs. tonic). Also these differences with respect to
response mode were well captured using a relatively minimal IFB neuron
model. Because the behavior of this simple neuronal model closely
resembles our experimental data, it should prove useful for predicting
population activity.
Responsiveness
The responsiveness of relay neurons to temporal frequency
was markedly different between burst and tonic firing whether we measured the mean response (F0) or the
fundamental, modulated response component
(F1). In terms of temporal tuning,
burst firing was low-pass for higher values of stimulus intensity
(I1) but markedly band-pass for lower
values of I1 (Figs. 5 and 8,
A and B). In contrast to burst mode, the
responses in tonic mode was broadband for the stimulus frequencies and
intensity levels tested (see Figs. 5 and 9, A and
B). Thus the tonic responses to very low frequencies were
relatively robust. Our results for burst firing are consistent with in
vivo temporal tuning measurements derived from extracellular recordings
in anesthetized cats (Mukherjee and Kaplan 1995), where
retinogeniculate transmission during burst responses to visual
stimulation were described having a temporal band-pass characteristic
and during tonic firing were tuned more broadly. Also although
responses in burst mode rarely were observed at 10 Hz regardless of
I1, neuronal responses in tonic mode
routinely were observed at 30 and 100 Hz. This is qualitatively
consistent with an in vitro study of guinea pig thalamic neurons
(McCormick and Feeser 1990
) that reported a
high-frequency cutoff for repetitive burst responses near 10 Hz. The
higher value obtained in this case may reflect the different stimuli
used, i.e., square pulses versus sinusoids.
Another feature of the high-frequency cutoff in burst mode that we
observed was the attenuation rather than an abrupt cutoff of the
F0 and
F1 values at 3 Hz measured as
spikes/cycle (see Fig. 8B). This could arise from two
different properties that we observed in the experimental results.
First, in about half the neurons, we observed subharmonic responses at
3 Hz and this would result in a decrease in spikes/cycle when averaged
over a number of trials. Second, although individual bursts at lower frequencies (<3 Hz) evoked essentially the same number of action potentials, neurons that did not respond with subharmonic bursts at 3 Hz often exhibited response attenuation via a decrease in the number of
spikes/burst (Fig. 12D). Although the IFB model is too
idealized to account for subharmonic burst responses, it does reproduce
this reduction of spikes at higher frequencies (Fig. 12D).
Previous computational work also has indicated that, during repetitive
hyperpolarizing current pulses, IT
gradually decreases at higher frequencies (Wang et al.
1991) and thus likely would evoke fewer action potentials.
Burst firing in response to sinusoidal current injection thus is tuned to roughly 0.3-3 Hz, particularly for lower amplitudes of activation. The failure of burst responses to higher frequencies is due to the low-pass filter properties of the membrane that lead to attenuated voltage responses at high frequencies of injected current (this also applies to tonic responses) and to the time constant for deinactivation of IT, which imposes a relative refractory period for the low-threshold Ca2+ spike, thereby limiting the high-frequency response. The IFB model responding in burst mode reproduces the high-frequency attenuation of F0 and F1 (Figs. 10, A and B, and 12D). However, it should be noted that this success is due in part to the fact that these response measures do not distinguish between subharmonic burst responses and reduction of spike/burst at higher frequencies.
The attenuation of burst responses observed at lower frequencies with moderate I1 values is a result of a fairly constant number of action potentials per burst in conjunction with one burst per stimulus cycle. When the response is measured in spikes/second, this results in an apparent attenuation at lower frequencies. However, the lowest I1 values giving rise to a burst response at 1-3 Hz resulted in no response at 0.1 or 0.3 Hz in 6 of 11 cells tested (see Fig. 5). The origin of the low-frequency cutoff observed in some relay neurons can be understood by considering models that are biophysically more detailed than the IFB model. Such models can be tuned to conform to our experimental observation that many relay neurons do not burst repetitively in response to any steady level of hyperpolarizing applied current. When this is done, one discovers that the membrane potential must exceed both a voltage threshold and also a minimum rate of change of the membrane voltage (i.e., dV/dt) to activate IT. A limitation of the minimal IFB model is that it lacks this latter requirement, that is, the IFB model responds with a low-threshold spike no matter how slowly V is brought above the burst threshold, Vh.
Our experimental observation of band-pass relay neuron responses
(in units of spike/cycle) may be of particular interest for the
following reason. Fast EPSPs, such as those activated via ionotropic
receptors (Conn and Pin 1997; Pin and Duvoisin
1995
; Recasens and Vignes 1995
), create a large
enough dV/dt to activate IT reliably. Much slower EPSPs, such
as those activated via metabotropic receptors (Conn and Pin
1997
; Pin and Duvoisin 1995
; Recasens and
Vignes 1995
), might be so slow that the cell may depolarize without activating IT. This also
raises the possibility that EPSPs activated by metabotropic receptors
could bring the cell from a hyperpolarized, burst-capable voltage
regime to tonic firing mode by moving
IT from the deinactivated state to
inactivated without ever activating
IT. If this metabotropic-based
depolarization was strong enough, tonic firing would be evoked because
there is no low-frequency attenuation for responses in tonic mode. We know that retinal inputs activate only ionotropic glutamate receptors on relay cells (Godwin et al. 1996
; McCormick and
von Krosigk 1992
) and are thus likely to activate
IT reliably when the relay cell is in
burst mode. Both cortical and brain stem modulatory inputs activate
ionotropic and metabotropic receptors (Godwin et al.
1996
; McCormick and von Krosigk 1992
; reviewed
in Sherman and Guillery 1996
), which raises the
intriguing possibility that activation of metabotropic receptors via
these inputs can change the relay cell's firing mode without
activating the cell.
Phase of response
During burst firing, both the experimental data and the IFB model
show an overall phase advance that lags slightly with increasing temporal frequency (Figs. 8B and 10B). Because
burst responses tend to be focused in phase, the quantitative value of
P1 is largely determined by the phase
of burst onset. The phase advance exhibited by the IFB model at 0.1 Hz
reflects the fact that V passes over the burst threshold,
Vh, during the upswing of the
sinusoidal current injection; however, the quantitative value
(P1 = 0.2 cycles) is sensitive to both
I1 and
I0 (e.g., the burst occurs earlier and
is consequently more phase advanced with greater
I0). The decrease in phase advance
with increasing frequency reflects the passive membrane properties in
the subthreshold regime (V < Vh), that is, V can be delayed
90° with respect to the applied current when the IFB model is
stimulated at higher frequencies (see Eqs. A2 and A3 in APPENDIX).
Tonic firing is different because there is less of a phase advance in
the experimental data with a slight lag with increasing frequency and
virtually no advance or change with frequency in the model. The reduced
advance compared with burst firing can be explained by the fact that
tonic firing provides a sustained response that will more symmetrically
distribute around the peak of the sinusoidal stimulus. The subtle
difference between the phase advance seen in tonic firing
experimentally versus none in the model can be explained by the modest
spike frequency adaptation (Vergara et al. 1998;
Wang 1998
) seen in geniculate relay cells (Smith
et al. 1999
). Spike frequency adaptation means that the response to a sustained input gradually reduces with time, and the
result with a sinusoidal stimulus is that the responses will be
slightly stronger before the peak of the sinusoidal stimulus than
after. This, in turn, is reflected in a slight phase advance. Because
the IFB model does not have spike frequency adaptation, it does not
show this phase advance for tonic firing.
Linear summation of responses
The index of nonlinearity also differed greatly between the two
firing modes, at least for low stimulus frequencies (0.1-1 Hz). Here
tonic responses showed excellent linear summation, whereas burst firing
did not, and this difference was captured effectively by the IFB model
(Figs. 8D-11D). Our analysis indicates that this difference in linearity at lower temporal frequencies is largely due to
the greater rectification of burst than of tonic responses. With
increasing frequency of stimulation, the nonlinearity index during
tonic mode became very similar to burst mode. However, here our
analysis indicates that this increase in nonlinearity for tonic firing
is a result not of increased rectification of the response but rather
of phase locking of the response to the stimulus. This is curious
because analogous measures of linearity of responses of geniculate
cells to visual stimuli recorded in vivo indicate that tonic responses
are reliably more linear than are burst responses at temporal
frequencies that includes the range tested here (Guido et al.
1992, 1995
; Mukherjee and Kaplan 1995
). Our IFB
model suggests an answer to this conundrum: when sufficient current
noise is added to the neuron model, the phase locking seen for
3 Hz
is largely eliminated, and tonic responses remain much more linear than
burst responses. Similar effects have been observed in vivo during
current injections of regular-spiking cells of the visual cortex
(Carandini et al.1996
) Because there is clearly more
noise, such as synaptic noise, during in vivo recording than the sort
of in vitro recording represented in Fig. 9D, we would
expect a reduction in phase locking during tonic firing in vivo.
However, it is interesting in this context to note that Reich et
al. (1997)
do find evidence for some phase-locking in responses
to drifting sinusoidal gratings during in vivo recording in the cat
lateral geniculate nucleus, but the phase locking was seen only to the
highest contrast gratings used.
IFB model
An integral part of this work is the construction and tuning of a
minimal model that quantitatively reproduces salient features of the
thalamic relay cell firing patterns, including the burst/tonic differences discussed in the preceding text. The conceptual novelty of
the IFB model is the compact account of both burst and tonic response
properties of relay cells in terms of two thresholds, Vh and
V, responsible for the activation
of burst and tonic spiking, respectively. When responding in tonic
mode, the IFB model is essentially a classical leaky integrate-and-fire
neuron model. When responding in burst mode, the dynamics of a single slow variable, h, characterize the inactivation and
deinactivation of IT that subserves
burst firing. Unlike more complicated thalamic relay cell models, the
IFB model is highly constrained to fit the Fourier analysis of
experimental responses presented here.
It is remarkable that most of the Fourier analysis of experimental data
is captured by the dual threshold character of the IFB model, and
particularly so is the ability to quantitatively fit responses of cat
geniculate relay cells with such a minimal characterization of
IT. Our observation that low-frequency
injected current often fails to activate
IT even though higher-frequency stimulation is successful as well as the observation that many neurons
do not repetitively burst for any range of injected current, together
suggest a relatively small, if any, window current in these neurons
(Coulter et al.1989). However, we have not explicitly probed for membrane potential bistability or other phenomenon that
might be induced by a window current (Williams et al.
1997
). Our success with the Fourier analysis using only a
caricature of IT is presumably due in
part to the use of a (relatively simple) sinusoidal stimulus and our
focus on neuronal responses at the (relatively coarse) level of PSTHs.
Thus it will be important to explore the generalizability of the model
by comparing experimental and theoretical responses to more complex
stimuli. Similarly, although here we found it unnecessary to include in
the IFB model the hyperpolarization-activated cation conductance,
Ih, in other contexts this conductance is
known to play a significant role (McCormick and Pape
1990a
).
Although the IFB model reproduces subharmonic tonic responses of relay neurons, it does not reproduce the subharmonic bursts occasionally observed experimentally. An important question is what aspect of the simplicity of the IFB model is responsible for this limitation. When relay cells are responding with subharmonic bursts, often a perfunctory low-threshold spike occurs on the crest of the cycle directly preceding a robust burst. Thus the relay cell appears to accumulate deinactivation of IT (i.e., h gradually increases) during missed cycles (i.e., those in which bursts responses do not occur). This suggests that the IFB model does not reproduce subharmonics in burst mode because the simplified kinetics of IT used do not allow the gradual increase in h.
To test this idea, we looked for subharmonic responses in a
mathematical model of the low-threshold spike that uses continuous activation and inactivation curves for IT that
are biophysically more realistic than the discontinuous curves used in
the IFB model (Wang et al. 1991). In agreement with our
experimental observations, we chose parameters for this model such that
repetitive low-threshold spikes (superharmonics) were not expressed for
any range of applied current. In a narrow frequency range, this
low-threshold spike model exhibited subharmonic responses similar to
those of the relay neurons presented here. In addition, in the
low-threshold spike model, deinactivation (h) does indeed
accumulate during missed cycles. Thus the inability of the IFB model to
reproduce the subharmonic burst responses observed occasionally in
experiments may be due to its minimal characterization of
IT. On the other hand, reproducing
subharmonic burst responses with a mathematical model that lacks
Ih does not eliminate the possibility that
Ih contributes to the experimentally
observed responses. Indeed, during periodic square pulse stimulation,
relay neuron models that include both
IT and Ih
can produce subharmonic bursts that appear to originate from the
temporal integration of hyperpolarization by
Ih (Wang 1994
).
Conclusions
There are two broad conclusions we can draw from these
studies. First, on the experimental side, we have used Fourier
techniques to characterize the input/output properties of geniculate
relay cells in tonic and burst firing mode. Among the findings was
evidence that tonic firing showed low-pass and sometimes broadband
temporal tuning, while burst firing was band-pass, peaking near 3 Hz
and often not responding to stimuli administered at lower frequencies (e.g., 0.1 Hz). These data are consistent with in vivo observations based on receptive field properties (Mukherjee and Kaplan
1995) and suggest that burst firing is not very responsive to
steady-state properties of the visual world but rather require temporal
change. We also found that, for frequencies <3 Hz, tonic responses
were more linear than burst responses, whereas phase-locking of tonic responses
3 Hz increased the nonlinearity of tonic responses to
levels near those seen during burst firing. Studies of receptive field
properties in vivo find, on the one hand, that tonic firing is
generally more linear than is burst firing for geniculate cells (Guido et al. 1992
, 1995
; Mukherjee and Kaplan
1995
) and, on the other hand, that phase locking can occur for
very high contrast stimuli (Reich et al.1997
). However,
the relationships among phase locking, linearity, temporal frequency,
and burst versus tonic firing mode in these in vivo studies has not
been exhaustively explored. Nonetheless it seems clear that many of the
receptive field properties noted above are at least partly due to the
membrane properties of the geniculate relay cells themselves,
particularly with respect to IT.
Second, on the theoretical side, we have been able to reproduce the salient features of relay neuron responses to sinusoidal input with a minimal IFB neuron model. The computational simplicity of this model makes it a good starting point for network simulations of retinogeniculate transmission that include important aspects of thalamic circuitry (e.g., GABAergic inhibition from local interneurons and neurons of the thalamic recticular nucleus). One of the keys to understanding thalamic relay function will be to understand how thalamic circuitry and corticogeniculate feedback controls the burst and tonic firing modes of geniculate relay neurons.
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APPENDIX |
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Current-frequency relation for the IFB model in tonic mode
When responding in tonic mode, the IFB model is essentially a
classical leaky integrate-and-fire neuron model (Knight
1972). In the absence of bursting, h eventually
decays to zero and, because it is inactivated, the low-threshold
Ca2+ current,
IT, will not contribute to the
dynamics of the membrane potential, V, which are effectively
given by
![]() |
(A1) |
Until V achieves the threshold for firing an action
potential, V, the membrane
potential of the model neuron is given by Eq. A1, solutions
of which take the form
![]() |
(A2) |
![]() |
(A3) |
![]() |
(A4) |
![]() |
(A5) |
Estimates of cutoff frequency in tonic and burst mode
Until either the tonic or burst threshold is crossed, V will be given by Eq. A2. If neither threshold is crossed for sufficiently long time, V will be given by V(t) = W(t) with W(t) as in Eq. A3. This solution represents the membrane potential oscillating under the influence of the sinusoidal applied current. Such responses were seen both experimentally and in simulations (Fig. 3, A and B, bottom). In the IFB model calculation in Fig. 3B, left, V is oscillating between the burst and tonic thresholds (i.e., in the gap region), whereas in Fig. 3B, right, V is oscillating below the burst threshold.
When the model is in the gap region,
Vh < V < V. Because
V(t) = W(t) and
W(t) is dependent on frequency, it is possible to
ask how much the frequency would have to be lowered in order for a
tonic or a burst response to be evoked by the model. This certainly
will not occur unless the maximum membrane potential exceeds
V
(evoking a tonic spike) or the
minimum membrane potential passes below Vh
(required for a burst response). Focusing on the first possibility, we
can find the maximum of V(t) = W(t) by differentiating Eq. A3 to give
![]() |
(A6) |
![]() |
(A7) |
![]() |
(A8) |
![]() |
(A9) |
![]() |
(A10) |
As a final point, notice that if we know the stimulus frequency,
f, we can solve for the values of
I0 and
I1 necessary to elicit a bursts
response. We arrive at an expression similar to Eq. A10
![]() |
(A11) |
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(A12) |
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ACKNOWLEDGMENTS |
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We thank Daniel Tranchina for helpful suggestions.
This research was supported in part by National Eye Institute Grant EY-03038 to S. M. Sherman. G. D. Smith was supported by a National Institute of Health Intramural Research Training Assistantship and National Research Service Award EY-06903-01.
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FOOTNOTES |
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Address reprint requests to S. M. Sherman.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 13 April 1999; accepted in final form 17 August 1999.
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REFERENCES |
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