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The Journal of Neurophysiology Vol. 88 No. 2 August 2002, pp. 761-770
Copyright ©2002 by the American Physiological Society
1Department of Neurobiology, Weizmann Institute of Science, Rehovot 76100; and 2Center for Neural Computation, The Hebrew University, Jerusalem 91904, Israel
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ABSTRACT |
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 TOP
 ABSTRACT
 INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Fuhrmann, Galit, Henry Markram, and Misha Tsodyks. Spike Frequency Adaptation and Neocortical Rhythms. J. Neurophysiol. 88: 761-770, 2002. Spike-frequency adaptation in neocortical pyramidal neurons was examined using the whole cell patch-clamp technique and a phenomenological model of neuronal activity. Noisy current was injected to reproduce the irregular firing typically observed under in vivo conditions. The response was quantified by computing the poststimulus histogram (PSTH). To simulate the spiking activity of a pyramidal neuron, we considered an integrate-and-fire model to which an adaptation current was added. A simplified model for the mean firing rate of an adapting neuron under noisy conditions is also presented. The mean firing rate model provides a good fit to both experimental and simulation PSTHs and may therefore be used to study the response characteristics of adapting neurons to various input currents. The models enable identification of the relevant parameters of adaptation that determine the shape of the PSTH and allow the computation of the response to any change in injected current. The results suggest that spike frequency adaptation determines a preferred frequency of stimulation for which the phase delay of a neuron's activity relative to an oscillatory input is zero. Simulations show that the preferred frequency of single neurons dictates the frequency of emergent population rhythms in large networks of adapting neurons. Adaptation could therefore be one of the crucial factors in setting the frequency of population rhythms in the neocortex.
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INTRODUCTION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Spike-frequency adaptation (SFA)
is the decrease in instantaneous discharge rate during a sustained
current injection and is a specialized feature of many types of
neurons. SFA has been observed in neurons of various systems from
several species, including non mammalian ones such as the crayfish
stretch receptor (Michaelis and Chaplain 1975
). In
mammals, SFA has been observed in rodent motoneurons (Granit et
al. 1963; Sawczuk et al. 1995), hippocampal CA1
pyramidal cells (Lancester and Nicoll 1987;
Madison and Nicoll 1984), and pyramidal cells of the
piriform cortex (Barkai and Hasselmo 1994). In the
neocortex of rodents, SFA has been identified in most pyramidal
neurons, in particular those that have been traditionally classified as
regular spiking cells, but in some bursting neurons as well
(Connors and Gutnick 1990; Mason and Larkman
1990; McCormick et al. 1985). SFA has also been
identified in neurons of other mammalian systems, including the rabbit
CA1 and CA3 pyramidal neurons (Moyer et al. 1996;
Thompson et al. 1996), cat motoneurons (Granit et
al. 1963; Kernell and Monster 1982) and layer V
neurons of the sensorimotor cortex recorded in vitro (Schwindt
et al. 1988; Stafstrom et al. 1984), neurons of
cat visual cortex in vivo (Ahmed et al. 1998), and even
regular spiking cells of the human neocortex (Avoli and Olivier
1989; Foehring et al. 1991; Lorenzon and
Foehring 1992).
The biophysical mechanisms underlying SFA are not yet established. SFA
has most commonly been linked to the phenomena of
afterhyperpolarization (AHP), found to follow current-induced
repetitive firing (Madison and Nicoll 1984;
Schwindt et al. 1988). AHP is an
action-potential-dependent hyperpolarized potential that markedly
summates with successive spikes. Because the build-up of AHP with
successive spikes is relatively slow, its effects on discharge
frequency are greater at later inter-spike intervals (Madison
and Nicoll 1984). The ionic mechanisms underlying AHPs and
their functions have been studied in neurons from several species, such
as the cat (Schwindt et al. 1988; Stafstrom et
al. 1984), guinea pig (Connors et al. 1982;
McCormick et al. 1985), and rat (Madison and
Nicoll 1984), and have been suggested to be largely produced by
Ca2+-activated slow K+
currents (Connors et al. 1982; Hotson and Prince
1980; Madison and Nicoll 1984; Schwindt
et al. 1988). In slices of human cortical tissue, AHP was
observed, and the currents underlying the medium and slow AHPs were
shown to influence the interspike interval during repetitive firing and
to produce SFA (Avoli et al. 1994; Lorenzon and
Foehring 1992). Other ionic currents that have been suggested
to contribute to the development of SFA include the M-current
(IM), which is a slow-activating
noninactivating voltage-sensitive potassium current (Madison and
Nicoll 1984; McCormick et al. 1993), and the
slow Na inactivation (Michaelis and Chaplain 1975;
Schwindt and Crill 1982).
The functional significance of SFA is not clear either. Some possible
roles of SFA have been suggested. These include the phenomena of
forward masking and selective attention
(Liu and Wang 2001; Wang 1998). In
forward masking, when two or more inputs are presented sequentially in
time, the neuronal response to the first input inhibits responses to
subsequent inputs by activating IAHP
with a delay. In selective attention, in the presence of two or more
inputs, the adaptation process can selectively suppress the neuronal
responses to weaker inputs so that the response to the strongest input
"pops-out" in time.
In this work, we studied the significance of SFA in modulating the
input-output properties of a neocortical neuron embedded in a noisy
environment. In particular, the response properties of an adapting
neuron to oscillatory inputs suggest a role for SFA in the
synchronization of neuronal assemblies. Previous results suggested a
role for adaptation in stabilizing synchroneous behavior (Crook
et al. 1998; van Vreeswijk and Hansel 2001).
Here we show that by knowing the characteristics of SFA in individual
neurons of the population, it is possible to predict the frequency of oscillations for which synchronization could occur, i.e., the frequency
of possible spontaneously emerging population rhythms.
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METHODS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Experimental
Slice preparation and recording procedures as in Markram
et al. (1997). Briefly, Wistar rats (13-15 days) were rapidly
decapitated, and neocortical slices (sagittal; 300 mm thick) were
sectioned (DSK, Microslicer, Japan). Neurons in the somatosensory
cortex were identified using IR-DIC video-microscopy (Zeiss Axioplan, fitted with [mult]40-W/0.75 NA objective; Zeiss, Oberkochen,
Germany), and patch-clamp recordings were obtained. Recorded neurons
were selected up to 120 µm below the surface of the slice and
separated from each other by up to 150 µm. Experiments were performed
at 32-34°C with extracellular solution that contained (in mM) 125 NaCl, 2.5 KCl, 25 glucose, 25 NaHCO3, 1.25 NaH2PO4, 1.5 mM CaCl2, and
1.5 mM MgCl2. Somatic whole cell recordings
(10-20 M access resistance) were obtained, and signals were amplified
using Axoclamp-2B amplifiers (Axon Instruments), captured on the
computer using pulse control (by Dr. R. Bookman and colleagues, Miami
University), and analyzed in programs written in Igor (Igor
Wavemetrics, Lake Oswego, OR). Pipettes solution contained (in mM) 100 K-gluconate, 20 KCl, 4 ATP-Mg, 10 phosphocreatine, 0.3 GTP, 10 HEPES,
and 0.5% biocytin (pH 7.3, 310 mOsm).
Modeling
Two models are used in this study. The first is an integrate-and-fire model of a spiking adapting neuron, used for detailed simulations. The second is a mean firing-rate model of an adapting neuron.
INTEGRATE-AND-FIRE MODEL OF AN ADAPTING NEURON.
To simulate the spiking behavior of a pyramidal neuron exhibiting SFA,
we consider a model of an adapting neuron, based on the classical leaky
integrate-and-fire model of a neuron (Tuckwell 1988).
The model for an adapting neuron takes into account an additional
hyperpolarizing potassium current, referred to as the adaptation
current (Treves 1993).
, the membrane potential of the neuron evolves
according to the following differential equation
|
(1) |
m is the membrane's time constant.
Isyn is the synaptic current, which is non-zero in cases
where the neuron is embedded in a network, and Iext is an
external current representing inputs from other brain areas.
Whenever the membrane potential V reaches threshold, a spike
is emitted and V is instantaneously set to a constant
resetting value (Vreset).
In most of the analysis we study the neuron's response to noisy
currents. We therefore consider
|
(2) |
I
(t) is an uncorrelated white noise with unit variance. We use the parameter
I, which is in picoampere units, to
characterize the amplitude of the noise.
We also consider an alternative, conductance-based model, in which
external inputs cause changes in the membrane's conductance, rather
than simple current injections. According to this model
|
(3) |
g
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(4) |
|
(5) |
is a
constant determining the step increase in n that occurs for
each spike, and N is the time constant for deactivation of the adaptation current. Thus the adaptation current
tends to accumulate when the neuron fires and decays between spikes
with the time constant of N. This
current is a phenomenological current and does not aim at describing a
specific biophysical mechanism but could be related to the outward
calcium-activated potassium current
(IAHP).
In this study, simulations were performed in the parameter regime where
n is significantly smaller than 1. Therefore to simplify the
analysis, the term (1
n) could be dropped off Eq. 5 without changing the qualitative behavior of the model.
MEAN FIRING RATE MODEL OF AN ADAPTING NEURON. To enable analytical calculations, a simplified model was constructed to simulate the mean instantaneous firing rate of an adapting neuron under noisy conditions, averaged over many repetitions of the same input. The model is based on two differential equations for the mean firing rate of the neuron and for the mean adaptation current, and a third analytical equation that translates mean voltages into mean firing rates.
The dynamics of the mean adaptation current, µa, is derived from the integrate-and-fire model of an adapting neuron by averaging over Eq. 5 for the case of Poisson firing statistics, with rate E
|
(6) |
Vk), assuming that the voltage for
these current injections remains close to threshold.
To approximate the dynamics of the mean firing rate (E) of
the neuron, we use a standard formulation, first presented in
Wilson and Cowan (1972). The model was developed for the
mean firing rate averaged over a population of neurons. Here we apply
it to describe the dynamics of the instantaneous firing rate of a
single neuron, averaged over many responses to different realizations of the same average input
|
(7) |
e is the
time constant underlying the dynamics of the mean firing rate. In the
case of white noise input statistics, the parameter
e is mainly controlled by the membrane time
constant, but its numerical value depends on parameters of the input,
such as its mean amplitude. This differential equation approximates the
evolution of E toward its stationary value in case of a
constant input, which is given by a function
(Rin · I) with
= Rin · I
|
(8) |
' = Vrest, H = Vreset . For µ above threshold, the function
(µ) is approximately linear and may be
written as (µ) =
b1 µ + b0, thus
simplifying subsequent analytical analysis.
Although Eq. 7 cannot be rigorously derived from the
detailed integrate-and-fire model, and although it was shown not to
accurately describe the firing rate dynamics (Gerstner
2000), it can still provide useful insights for understanding
the neuron's behavior.
SIMULATIONS OF NETWORKS COMPOSED OF ADAPTING NEURONS.
We consider small networks of only a few neurons, with manually
designed architecture of connections as well as large scale randomly
connected networks. The activity of the networks, composed of adapting
neurons, is simulated using the detailed integrate-and-fire model
formulated in Eqs. 1 and 5. All neurons can be
driven by an externally injected current (Iext) as well as
by synaptic current (Isyn), which is induced by the activity
of the presynaptic neurons. For each simulated neuron, once a spike
occurs, a synaptic current is activated in all of its postsynaptic
targets. Synaptic currents (EPSCs) are simulated by the difference of
two exponentials, multiplied by the synaptic strength, J
|
(9) |
1 and 2
are the rise and decay time constants of EPSCs and
Vsyn is the synaptic reversal
potential in use.
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RESULTS |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Response properties: comparison between experiments and models
To compare the performance of the model neuron with that of real
neurons, similar current injections were applied to both model neurons
and also to several layer II-IV pyramidal neurons, whose voltage
responses were recorded in slice preparations of rat somatosensory
cortex. Under in vivo conditions, however, a neocortical neuron is
exposed to the background activity of its presynaptic neurons and thus
experiences noisy input currents (Softky and Koch 1993).
This study, therefore focuses on the the response of an adapting neuron
to various noisy current injections.
RESPONSE TO NOISELESS STEP CURRENTS. The response of the neurons to noiseless step currents was used primarily to compare the response of the model neuron to that of real neocortical neurons and to extract realistic model parameters for subsequent analysis. In Fig. 1A, we show the typical response of an adapting neuron to a constant current injection. It can be seen that the interval between subsequent spikes in the train is increasing.
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) and therefore was not considered as part of the spike frequency adapting behavior of the
studied neurons. In some traces, a slower process followed the early
phase and caused an additional increase in ISI. All three components
were previously observed by Kernell and Monster (1982)
and Sawczuk et al. (1995) and were referred to as the
initial, early and late phases of
adaptation. The construction of the model aimed at simulating the early
phase of adaptation only.
The experimental traces were fit using the model by searching for the
most appropriate set of model parameters (,
N)
that approximates the experimental results. Using the chosen parameters, it was possible to replicate the evolution of ISIs within
the train (Fig. 1B, solid line). The experimental first ISI
is significantly shorter than that of the model, supporting the
possibility that it may be caused by a different mechanism, perhaps
related to the bursting behavior of these neurons (McCormick et
al. 1985). In other traces with the same model parameters but a
different amplitude of current injected, a good match to the responses
of the same neuron was also obtained for all ISIs (not shown),
excluding the first few, further supporting the possibility of a
different underlying mechanism for the first ISIs.
RESPONSE TO NOISY STEP CURRENTS. Under in vivo conditions, where spontaneous activity of presynaptic neurons causes strong background fluctuations in the membrane potential, non noisy step currents represent unrealistic inputs to a neocortical neuron. We therefore explored the behavior of the adapting neuron in response to a fluctuating input.
Figure 2A illustrates an example of voltage responses of a model neuron to a step current with added white noise. Apparently, in the case of noisy injections, adaptation cannot be characterized simply as an increase in the length of subsequent ISIs. In this case, the response is characterized using the poststimulus time histogram (PSTH) of the neuron. In Fig. 2B, the PSTH of a model adapting neuron for 500 different realizations of a noisy step current is shown. In each trial, the noise realization was different. There is a peak in firing rate at the beginning of the injection interval (preceded by a short latency), such that the probability of the neuron firing is higher at the beginning of the trace, and then decays to a lower steady-state value.
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RESPONSE TO NOISY OSCILLATORY CURRENTS.
The response of the model neuron to oscillatory currents of the form
I(t) = I0 + I1 sin(2
ft), with white
noise, was examined. An example of an adapting neuron's response to
such an injected current is illustrated in Fig.
3A (the initial transient is
not shown). Note that the oscillatory response has the same frequency as that of the injected current. However, in the presented example, the
phase of the response is advanced relative to that of the current by
almost 40°.
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Hz) is the frequency at which
no phase shift occurs. We define to be the preferred
frequency of the neuron or the zero phase frequency.
Note that the firing rate of the neuron, and thus the number of spikes
per cycle, is also affected by the frequency of the input current, and
there exists also a special frequency for which the gain of the
response, in terms of the average firing rate, is maximal. However, we
stress that in the context of this study we define the preferred
frequency of the neuron to be the frequency at which the phase
shift between the discharge rate of the adapting neuron and its input
current is zero.
The existence of this property of an adapting neuron can be explained
by the interplay between two opposing mechanisms. On one hand, the time
constant of the firing rate dynamics (e) tends to cause a delay in the response of the neuron to its injected current
(Eq. 7). This mechanism dominates at high-input frequencies. On the other hand, the dynamics of the adaptation current
Ia (Eq. 6) tends to advance
the phase of the mean firing rate. This is prominent at lower input
frequencies and is due to the accumulation of adaptation current during
the rising phase of the discharge rate, which induces an advanced
decline of the firing rate. The preferred frequency, where zero phase
shift is obtained, is reached when the two opposing mechanisms balance
each other. Note that at very low frequencies, the phase shift
approaches zero again because Eqs. 6 and 7 are
effectively at their steady-state solutions and the firing rate follows
the oscillations of the input.
Simulations of the conductance based model of an adapting neuron
(Eq. 3) result in similar phase-frequency curves, with a preferred frequency of zero phase shift. For parameters chosen such
that the mean input current is the same as that of the corresponding current-based neuron, the preferred frequency changes only slightly, even when the average input conductance increases up to 200% from its
initial value, as reported to occur under in vivo conditions (Borg-Graham et al. 1998). Therefore for simplicity,
further analysis was performed for the current-based neuron.
To gain insight on the dependence of the preferred frequency, , on
the parameters of adaptation, we used the simplified model for the mean
firing rate, to analytically compute , under a linear approximation
for small amplitudes of oscillations
|
(10) |
, N in the model), which determine the degree
and time course of adaptation. Experimental studies have shown that neuromodulators, such as ACh, can reduce the degree of adaptation (Tang et al. 1997) e.g., by reducing
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is not a constant value dictated by the parameters of the neuron alone but is also modulated by the parameters of the input current. In particular, increasing the mean value of the
input current (I0) causes an increase
in (Fig. 6A). In the
framework of the firing rate model of an adapting neuron, this
dependence is related to the fact that the time constant underlying the
dynamics of the mean firing rate (e) depends
on the parameters of the input current (Holt 1998).
Increasing I0 pushes the voltage
closer to the firing threshold, resulting in faster dynamics of the
discharge response, i.e., a shorter e. Therefore increasing I0 causes
e to decrease, which in turn determines a
higher preferred frequency (Eq. 10).
Increasing the amplitude of current oscillations
(I1) induces lower preferred
frequencies (Fig. 6B). However, this dependence is much
weaker than the dependence on the mean value of the input current
(I0).
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Emergence of population rhythms in networks of adapting neurons
To study the implications of the preferred frequency on synchronization in neural networks, small networks of model neurons, driven by an external oscillatory input, were considered. Synchronous oscillations of discharge rate are achieved if all neurons exhibit the same phase shift of discharge, relative to the external input current, for a given modulation frequency.
In Fig. 7, results are presented for a network of four identical neurons, in which one neuron receives an oscillatory external input and the others receive a constant current injection (DC shift) to keep them firing at the same mean discharge rate. Because the neurons are identical, they all have the same phase-frequency curve relative to their own input current (Fig. 7A). We therefore expected the neurons to synchronize at their preferred frequency. However, synchronization was achieved at a lower frequency than the "preferred frequency" of the neurons (Fig. 7). The neurons synchronize at a frequency in which they all produce a phase shift that exactly opposes the phase shift caused by the synaptic delay at that frequency. We denote this frequency as the corrected preferred frequency and suggest the following method to determine its value. For a given frequency of input current, the phase shift of each neuronal element relative to its own input and the phase of the synaptic delay is determined. The "corrected" phase shift (Fig. 7B) is obtained by adding the phase of the synaptic delay to the phase shift of the response. The frequency for which the `corrected' phase shift is zero is the frequency of current injection for which the neurons will exhibit synchronization. This can be seen in Fig. 7C, where at the corrected preferred frequency, the phase shift of the rate response relative to the externally injected current is shown to be equal for all neurons. We therefore expect that the corrected preferred frequency will determine the rhythms in neocortical networks.
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This prediction was tested in a large, randomly connected, homogenous network of 200 identical adapting neurons, each receiving an uncorrelated noisy current injection with a constant mean, to induce spontaneous firing (Fig. 8). Under conditions in which the average firing rate of the neurons was well above zero at all times, a synchronized oscillatory spiking activity appeared spontaneously in the network, for sufficiently strong synaptic connectivity. Note that no external oscillatory current was injected to any of the neurons in the network, and therefore the population rhythm is an emergent property of the network. In the presented example (Fig. 8, A and B), the frequency of the synchronized oscillations is 6.64 Hz. The frequency of the population rhythm remains constant over time. However, the amplitude of modulation in the average population histogram slowly waxes and wanes. This is expected because each neuron in the population experiences a slightly different input current due to random connectivity and noisy current injection, and therefore the frequency of oscillations in subgroups of neurons may be slightly different, resulting in beats of the activity. The input current that each neuron in the population receives is a combination of the externally injected current, as well as the synaptic current. The mean input current (I0) and the amplitude of the oscillations of the input current (I1) that a representative neuron of the population received on average in the simulation were determined. This was used to construct the corrected phase-frequency curve for a single neuron in the homogenous network, taking into account the phase of the synaptic delay at each of the input frequencies (Fig. 8C). The corrected preferred frequency of the neuron was determined as 6.7 Hz. From this result, as well as from simulation results of similar networks with different model parameters, we conclude that the frequency of the emerging population rhythm in a large network is indeed predicted by the corrected preferred frequency of the single neurons.
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Exploration of various homogenous populations revealed that the
frequency of the population rhythm is indeed lower for networks composed of neurons with smaller
It could be argued that it is not realistic to assume that a
neocortical network consists of identical neurons. Rather, it is more
plausible that the parameters of the neurons are randomly distributed
across the population. We therefore explored the effect of having a
heterogeneous population of adapting neurons whose parameters are all
identical, except that
If the strength of the connections is increased, the network can switch
to a different regime of activity. This regime is characterized by
synchronous bursts of firing across the network, with zero activity
between the bursts, and was recently studied in noiseless networks of
adapting neurons (van Vreeswijk and Hansel 2001).
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DISCUSSION |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Spike-frequency adaptation in neocortical pyramidal neurons was
examined using the whole cell patch-clamp technique and
phenomenological models of neuronal activity. Noisy current was
injected to reproduce the irregular firing typically observed under in
vivo conditions (Softky and Koch 1993). Spike-frequency
adaptation was shown to play an important role in shaping the response
of a neocortical neuron to such noisy stimuli. A detailed model was
used to simulate the spiking response of the neocortical pyramidal
neuron as well as a simplified model that captures its response in
terms of the instantaneous average discharge rate.
In the intact brain a neuron is embedded in large, spontaneously active networks. It is therefore generally assumed that a neuron in vivo experiences noisy inputs due to the background activity of its presynaptic neurons. We simulate this background activity by adding white noise to the injected stimuli. The response of a neuron under such conditions, was quantified by computing the poststimulus time histogram (PSTH) of the neuron and thus reflects mean firing rate characteristics only.
The study of the output of a model adapting neuron exposed to
oscillatory inputs revealed a dependence of the firing rate response on
the frequency of stimulation. Low-frequency modulated inputs produced a
phase advance of the output response relative to the input current,
whereas high-frequency modulated stimuli induced a phase delay of the
response. A special frequency was observed, referred to as the
preferred frequency of stimulation, for which the phase
delay of neuron's activity relative to the input is zero. The
prediction of such phase-frequency curves was confirmed by recordings
of neocortical pyramidal neurons. Additional experimental support for
the existence of the predicted phase-frequency curves is obtained from
the firing rate responses of thalamic relay neurons in the cat's
lateral geniculate nucleus (Smith et al. 2000) and of
regular spiking neurons in the guinea pig visual cortex
(Carandini et al. 1996), both recorded in
brain-slice preparations. The latter study also supports the predicted
dependence of the phase-frequency curves of neocortical neurons on the
DC level of current injection, I0 (see
also Kamondi et al. 1998 for low-frequency modulation).
Recently, it was shown that the response of nonadapting
integrate-and-fire neurons to noisy oscillatory currents is also
sensitive to the statistical properties of the noise (Brunel et
al. 2001; Rudolph and Destexhe 2001). In
particular, if the noise is low-pass filtered, e.g., due to the finite
duration of synaptic currents, the dynamics of the firing rate becomes
very fast for high-frequency oscillations. Hence, the phase of the
delay approaches zero even in the absence of an adapting mechanism.
However, for realistic values of parameters, this deviation of behavior
from the white noise case occurs at frequencies that are significantly
above the range we focus on in this study.
The significance of the preferred frequency of an adapting neuron for the emergence of synchronous population rhythms in the neocortex was investigated. The study of small networks, composed of only four neurons, one of which was stimulated by an externally oscillatory noisy current, suggested that the neurons may synchronize at a corrected preferred frequency, which takes into account the phase of the synaptic delay. This prediction was tested in recurrent networks of 200 adapting neurons with no externally oscillatory current injection. Nevertheless an oscillatory synchronized population activity spontaneously emerged. We found that the frequency of the population rhythm was indeed predicted by the corrected preferred frequency of single neurons. We conclude that the frequency of population rhythms can be predicted by the parameters of single neocortical neurons.
According to the model analysis, the preferred frequency depends on the
parameters of adaptation and thus can be adjusted by neuromodulators,
such as ACh (Tang et al. 1997), that affect the degree
of adaptation of pyramidal neurons. High concentrations of
neuromodulators can result in the shut off of adaptation currents and
hence in the abolishment of population rhythms. Adaptation could
therefore be one of the crucial factors in setting the frequency of
population rhythms in the neocortex. Indeed, in Sanchez-Vives and McCormick (2000), it is suggested that the slow
oscillations in the neocortex are generated through a recurrent network
of excitatory connections and that the periodicity is affected largely by the time course of the outward currents generating the slow AHP of
pyramidal and spiny stellate cells. Moreover, it has been shown that
the low-frequency oscillations are indeed suppressed by a variety of
neurotransmitters, including ACh and norepinephrine (NE), presumably
through the reduction of specialized K+
conductances, such as that underlying the AHP currents (Steriade et al. 1993). Faster rhythms are possibly determined by other mechanisms, which depend on the activity of inhibitory interneurons, and could therefore be less sensitive to the effect of neuromodulators (Brunel 2000; Tsodyks et al. 1997;
Wilson and Cowan 1972).
In addition to the expected significance of adaptation in determining
possible frequencies for neocortical population rhythms, theoretical
models have also suggested that the phase-relation of spike occurrence
relative to the population cycle may be capable of carrying sensory
information (e.g., Buzsáki and Chrobak 1995; Hopfield 1995; Kamondi et al. 1998;
Laurent 1996; Lisman and Idiart 1995;
Tsodyks et al. 1996). Experimental support for such
"phase coding" has been specifically demonstrated by the phenomenon
of spike phase precession observed in CA1 pyramidal "place cells" (O'Keefe and Recce 1993; Skaggs et al.
1996). These place cells undergo progressive phase precession
during the time that the rat crosses the place field of the cell. In
this sense, the phase shift of the spike discharge relative to the
theta population cycle encodes the rat's position in space. It has
been suggested (Kamondi et al. 1998) that this phase
precession is explained by increasing depolarization due to increased
excitation by afferents in the center of the field, causing the cells
to fire progressively earlier during the theta cycle. Such a dependence
of the phase on the DC level (I0) for
low-frequency modulated inputs is in accordance with our observations.
In the context of `phase-coding' of sensory information, adaptation
might have a role in enriching the coding language, by adding negative
phase shifts to the vocabulary of the code.
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APPENDIX |
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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Derivation of the preferred frequency of an adapting neuron (Eq. 10)
Equation 10 was derived under the assumption that the
input current is in the linear range of
(µ). In this range,
(µ) can be approximated by
b0+ b1 µ. We
analyze the approximated mean firing rate model, formalized in
Eqs. 6 and 7. If the externally injected current
is oscillatory, it can be represented as
|
(A1) |
|
t;
µa = µa* + I1a
eit.
The derivatives with respect to t are
therefore dE/dt = i
E1 eit;
dµa/dt = i
I1a
eit. By inserting the
preceding terms into Eqs. 6 and 7, respectively, one can compute
|
(A2) |
N Imax.
E1 = |E1|ei
E is a
complex number. Its magnitude (|E1|) will
determine the amplitude of firing rate oscillations, and the phase
shift relative to the injected current (
E) is
given by its angle
|
(A3) |
|
(A4) |
|
(A5) |
E = 0, we can compute the that satisfies
this condition
|
(A6) |
Vk).
The preferred frequency of the neuron is = (
.
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ACKNOWLEDGMENTS |
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This work was supported by the Israeli Academy of Science, Office of Naval Research, Human Frontiers Science Program, and the Edith Blum Foundation (New York).
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FOOTNOTES |
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Address for reprint requests:M. Tsodyks, Dept. of Neurobiology, Weizmann Institute of Science, Rehovot 76100, Israel (E-mail: misha{at}weizmann.ac.il).
Received 16 October 2001; accepted in final form 4 March 2002.
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TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX
REFERENCES
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response of the mean firing rate model of an adapting neuron for the
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