 |
 Previous Article | Next Article 
The Journal of Neuroscience, October 1, 1998, 18(19):7972-7986
Active Membrane Properties and Signal Encoding in Graded
Potential Neurons
Juergen
Haag and
Alexander
Borst
Friedrich-Miescher-Laboratory of the Max-Planck-Society, D-72076
Tuebingen, Germany
 |
ABSTRACT |
We investigated the influence of active membrane properties on the
precision by which the stimulus velocity is encoded in the membrane
potential of a motion-sensitive interneuron in the blowfly. The
so-called HS-cells respond to visual motion stimuli with a graded shift
in membrane potential. Superimposed on this graded response are small
spike-like events. This "mixed" visual response mode can be
modified by current injection in two different ways. (1) By ongoing
injection of hyperpolarizing current, the spike-like events are turned
into full-blown action potentials, and (2) by injection of depolarizing
current, the spike-like events become completely suppressed. The visual
response then consists of a graded shift of membrane potential only. As
a measure of the fidelity, we calculated the coherence between the
motion stimulus and the response of the cell elicited with different
electrical manipulations of the cell. We found that the coherence was
highest for the cell at rest. Any electrical manipulation resulted in a
reduced coherence. This was attributable partly to a lower
signal-to-noise ratio and partly to an increased nonlinearity in the
response. By applying a threshold operation we transformed the analog
membrane response into an all-or-none spike train. A comparison between these two ways of signal representation revealed that more information about the stimulus velocity is inherent in the analog membrane potential than in the spike train.
Key words:
Key Words: neural coding; reverse reconstruction; graded potential
neurons; active membrane properties; motion detection; reliability
| |
INTRODUCTION |
Most neurons communicate with each
other by sending trains of action potentials along their axons.
However, in addition to these classical spiking neurons, another type
of nerve cells called "graded potential neurons" is found in
various parts of the nervous system in vertebrates as well as in
invertebrates. In contrast to the former, graded potential neurons
usually do not produce regular action potentials but rather shift their
membrane potential in a graded way according to the prevalent input
signal.
A long-standing problem concerning these graded potential neurons is
the question of what difference exists between them and spiking neurons
with respect to the information they convey about the input signal
(Bullock, 1981 ). Is the graded mode more reliable than the spiking
mode? Can graded neurons carry more information than spiking ones or
the other way around? We examined this question previously by comparing
signal encoding in spiking (H1-cells) and graded potential neurons
(HS-cells) of the fly visual system (Haag and Borst, 1997). Both
neurons belong to the class of lobula plate tangential cells (LPTCs),
located in the posterior part of the third visual neuropile (lobula
plate) of the blowfly, which are known to respond to visual motion
stimuli in a directionally selective way (Borst and Egelhaaf, 1989,
1990; Egelhaaf et al., 1989). H1-cells communicate between the lobula
plates of both hemispheres by sending trains of action potentials along
their axons. HS-cells synapse onto descending neurons and respond to visual motion by a graded shift of their axonal membrane potential (Hausen, 1982a, 1982b, 1984; Borst and Haag, 1996). Applying the so-called "reverse reconstruction technique" (Bialek et al., 1991; Bialek and Rieke, 1992; Theunissen, 1993; Gabbiani et al., 1996; Theunissen et al., 1996), we showed that the time-course of image velocity could be better retrieved from the graded signals of HS-cells
than from the spike trains of H1-cells, the reason being the limited
dynamic range of the spiking neuron for inhibitory stimuli attributable
to the low spontaneous spike frequency (Haag and Borst, 1997). This
might lead to the conclusion that a purely graded neuron without any
active membrane properties could perform optimally in representing
sensory information. However, despite their graded response to visual
motion, HS-cells house various kinds of voltage-gated currents.
Voltage-clamp experiments revealed that these cells show a fast sodium
inward current that, depending on the resting state of their membrane
potential, can lead to spike-like events superimposed on the graded
shift of membrane potential (Borst and Haag, 1996; Haag et al., 1997).
These active processes have been previously shown to enhance the
cellular responses to high-frequency synaptic input signals (Haag and
Borst, 1996). By additional manipulation of the resting membrane
potential via injection of hyperpolarizing currents, these cells can be
turned from their normal "mixed" visual response mode into almost
purely spiking cells (Hengstenberg, 1997), or, alternatively, by
injection of depolarizing current, into purely graded cells (Haag and
Borst, 1996).
The analysis of the role of active membrane properties with respect to
the information they convey might be of further interest because the
dendrites of all neurons transmit signals mainly in a graded potential
manner, and, as in HS-cells, many of them house various voltage-gated
channels (Hirsch and Gilbert, 1991; Stuart and Sakmann, 1994; Yuste et
al., 1994; Callaway and Ross, 1995; Spruston et al., 1995). The
contribution of these active membrane properties to dendritic
information processing is not yet fully understood (Yuste and Tank,
1996).
In the present study we compare the encoding of velocity information in
HS-cells under various manipulations of their membrane potential and
ask to what degree fast membrane processes contribute to a more
accurate encoding as compared with a purely spiking or a purely graded
response mode. This study thus aims at a functional understanding of
the impact that active membrane processes have on neural coding in
graded potential neurons. We will also investigate to what degree the
usual resting potential of HS-cells represents an ideal set-point with
respect to the signal-to-noise levels inherent in the cellular membrane
signals.
| |
MATERIALS AND METHODS |
Preparation and set-up
Female blowflies (Calliphora erythrocephala) were
briefly anesthesized with CO2 and mounted ventral side up
with wax on a small preparation platform. The head capsule was opened
from behind; the trachea and airsacs that normally cover the lobula
plate were removed. To eliminate movements of the brain caused by
peristaltic contractions of the esophagus, the proboscis of the animal
was cut away, and the gut was pulled out. This allowed stable
intracellular recordings of up to 45 min. The fly was then mounted in
an upright position on a heavy recording table with the stimulus
monitors in front of the animal. The fly brain was viewed from behind
through a Zeiss dissection scope.
Stimulation
Stimuli were generated on Tektronix 608 monitors by an image
synthesizer (Picasso, Innisfree) and consisted of a one-dimensional grating of 14° spatial wavelength and 87% contrast displayed at a
frame rate of 200 Hz. The mean luminosity of the screen was 11.2 cd/m2. The intensity of the pattern was
square-wave-modulated along its horizontal axis. The angular width of
the stimulus fields was 40° in the horizontal and 28° in the
vertical direction as seen by the fly. To identify the cells by their
visual response properties, cells were first stimulated by the pattern
moving back and forth at 28°/sec. When the actual experiment was
started, the stimulus moved at a pseudo-random velocity with a flat
spectrum up to 30-40 Hz (Fig. 1). The
mean velocity of the stimulus was 0°/sec with a SD of 99.5°/sec.
One stimulus sweep lasted for 40 sec, and a variable number of sweeps
(5-10) were presented to each cell during one experiment. This was
repeated for each current injection that was imposed on the cell via
the recording electrode. Three levels were used, each of which turned
the cell into a distinct response mode: 0,  3, and +3 nA. The
injection of the hyperpolarizing current led to a potential shift of
approximately 12 to 14 mV; the injection of depolarizing current
led to a shift in membrane potential of approximately +10 mV.
View larger version (19K):
[in this window]
[in a new window]
|
Figure 1.
Amplitude spectrum of the stimulus used for all
experiments. The stimulus has a flat amplitude spectrum up to 30-40
Hz.
|
|
Recording
For intracellular recordings of HS-cells, electrodes were pulled
on a Brown-Flaming micropipette puller (P-97) using thin-wall glass
capillaries with an outer diameter of 1 mm (Clark, GC100TF-10). When
filled with 1 M KCl they had resistances of ~20 M . A
SEL10-amplifier (npi Electronics), which was operated in the bridge
mode, was used throughout the experiments. Out of the three different
HS-cells that are located in the lobula plate of each brain hemisphere (HSN-, HSE-, and HSS-cells) we recorded only from HSN- and HSE-cells. Because these cells, apart from their different receptive field locations, did not exhibit any differences in their response
properties, data from both cell types were pooled and are collectively
referred to as "HS-cells" in the following. All recordings were
made in the axons of these cells. Extracellular recordings of H1-cells were made using standard tungsten electrodes with a resistance of ~5
M. Extracellular signals were bandpass-filtered and subsequently processed by a threshold device delivering a 100 mV pulse of 1 msec
duration on each spike detected. For data analysis, the output signal
of the threshold device as well as the stimulus function controlling
the velocity of the pattern were fed to a PS/2 PC via a 12-bit A/D
converter (Metra Byte µCDAS-16G, Keithley Instruments) at a sampling
rate of 2 kHz and stored to hard disc.
Data evaluation
For theoretical background, also see Shannon and Weaver (1949),
Cover and Thomas (1991), Theunissen (1993), Theunissen et al. (1996),
and Rieke et al. (1997).
Reverse filter. Consider a stimulus
S(t) that causes, by some unknown transformation, a response
R(t). We want to estimate S(t) from
R(t). To optimize the reconstruction one chooses the filter
Grev, which minimize the mean square
error  2 between S(t) and
Sest(t):
|
(1)
|
Minimizing the mean square error leads to a filter
Grev. In the linear case the equation for this
optimal filter can be solved in frequency space (with   denoting
the average over different stretches of data Si
and Ri and * denoting the complex conjugate; see
Implementation below):
|
(2)
|
This filter represents, in frequency space, the average
cross-correlation between stimulus and response divided by the average auto-correlation of the response. It is the slope of a linear regression for Si-Ri
pairs at each frequency.
Coherence function. The following section
describes the calculation of the coherence function. On the basis of
the signal-to-noise ratios (SNRs), we will also define and calculate an
expected coherence function assuming that the system uses a linear
encoding scheme. For evaluating the quality of the reconstruction, we
calculate the gain relating S(t) to
Sest(t).
This gain is defined as:
|
(3)
|
Sest(f) relates to
R(f) and S(f) in the following way:
|
(4)
|
Inserting (4) into (3) yields:
|
(5)
|
This quantity, called the coherence  2, represents
the product of the average cross-correlations between the stimulus and the response and vice versa, divided by the stimulus and response power. It can be also understood as the product of the optimal linear
forward filter Gfwd transforming S
into R, and the optimal linear reverse filter
Grev.
It follows from Equation 5 that 0  2 1.
The deviation of a measured coherence from 1 can be attributed
to two different causes (1) the system is not linear; (2) the system is
corrupted by noise. To disentangle these two possible sources, we
calculated an expected coherence exp2
for a linear system.
Given a linear system that is corrupted by additive noise,
Ri(f) = Gfwd
Si(f) + Ni(f),
Equation 5 turns into:
If the stimulus and the noise are uncorrelated
Si(f)·Ni(f) = 0, for all frequencies (f) this expression simplifies
to:
|
(6)
|
The signal-to-noise ratio is defined as the quotient of the
signal and the noise power. Because the signal is the average response,
i.e., the response without noise, this expression becomes:
|
(7)
|
Combining Equations 6 and 7 yields:
|
(8)
|
Thus, based on the measured SNR, we can assign an expected
coherence exp2 to a system, assuming
that it is linear and the stimulus and noise are uncorrelated. The
expected coherence can be also written as the ratio of the power of the
average response and the average power of the response. Comparing this
expected with the actually measured coherence allows the difference
1 2 to split up into a part that is caused by
noise (1 exp2) and another
fraction that is caused by nonlinearity
(exp2 2).
Upper and lower bound of information rate. We will
now show how the two terms introduced above, i.e., the measured
coherence 2 and the coherence as expected for a linear
system, exp2, given a certain
signal-to-noise ratio SNR, relate to the lower and upper
bound of the information rate in a neural signal, respectively.
If the mean response and the noise have a Gaussian distribution and are
independent of each other, the upper bound or "channel capacity"
can be calculated as:
![<UP>Info<SUB>UB</SUB></UP>(<UP>bits</UP>/<UP>sec</UP>)=<LIM><OP>∫</OP><LL>f<UP>=</UP>0</LL><UL>f<UP>=</UP>k</UL></LIM><UP>log</UP><SUB>2</SUB>[1+SNR(f)]df.](/content/vol18/issue19/fulltext/7972/img011.gif)
|
(9)
|
By inserting Equation 8 into 9, one obtains:
![<UP>Info<SUB>UB</SUB></UP>(<UP>bits</UP>/<UP>sec</UP>)=<UP>−</UP><LIM><OP>∫</OP><LL>f<UP>=</UP>0</LL><UL>f<UP>=</UP>k</UL></LIM> <UP>log</UP><SUB>2</SUB>[1−&ggr;<SUP>2</SUP><SUB><UP>exp</UP></SUB>(f)]df.](/content/vol18/issue19/fulltext/7972/img012.gif)
|
(10)
|
We now calculate the lower bound of the information rate. This
calculation is based on the data processing inequality theorem, which
says that no clever manipulation of R can increase the
inference that can be made from R about S.
Given a processing chain S  R Sest, then
I(S, Sest) I(S, R).
Therefore we can safely transform the response R by
whatever filter and will never overestimate the information in
R about S. Hence it follows that the information
in Sest about S will be a
lower bound of the real information that is in R about
S. We define a signal-to-noise ratio of our reconstruction, SNRRec, as the mean power ratio of
Sest and the difference between
S and Sest:
|
(11)
|
This expression is identical to the one used by Rieke et al.
(1997) who defined SNRRec as the mean power
ratio of the stimulus and the difference between S and
Sest/2. Combining
Equations 4 and 5 yields:
|
(12)
|
If we use SNRRec instead of
SNR, Equation 9 turns into:
![<UP>Info<SUB>LB</SUB></UP>(<UP>bits</UP>/<UP>sec</UP>)=<LIM><OP>∫</OP><LL>f<UP>=</UP>0</LL><UL>f<UP>=</UP>k</UL></LIM> <UP>log</UP><SUB>2</SUB>[1+SNR<SUB><UP>Rec</UP></SUB>(f)]df.](/content/vol18/issue19/fulltext/7972/img015.gif)
|
(13)
|
By inserting Equation 12 into 13, the lower bound on the
information rate equals:
![<UP>Info<SUB>LB</SUB></UP>(<UP>bits</UP>/<UP>sec</UP>)=<UP>−</UP><LIM><OP>∫</OP><LL>f<UP>=</UP>0</LL><UL>f<UP>=</UP>k</UL></LIM> <UP>log</UP><SUB>2</SUB>[1−&ggr;<SUP>2</SUP>(f)]df.](/content/vol18/issue19/fulltext/7972/img016.gif)
|
(14)
|
Implementation. The signals were evaluated
off-line by a program written in Turbo-Pascal (Borland) using several
routines from Numerical Recipes (Press et al., 1988). Each continuous
40 sec stretch of the stimulus S(t) and response function
R(t) was cut into time segments of 4.096 sec duration
[Si(t) and
Ri(t)], respectively. This resulted
in nine Si(t) and
Ri(t) data stretches per sweep.
Because during an experiment 5-10 sweeps per stimulus condition were
recorded, 45-90 data segments for each cell and stimulus condition
were obtained. Each of these segments,
Si(t) and
Ri(t), was Fourier-transformed to
Si(f) and
Ri(f), and the cross- and
auto-correlations were estimated as the products (averaged over the
number of sweeps and the segments within one sweep) of the complex
functions. The coherence was calculated for the different level of
current injection for each cell and then averaged over different cells.
The signal and noise spectra were measured as follows. From the neural
signals obtained in response to repeated stimulus presentations, we
first calculated the mean response R(t). To calculate the
noise within each stimulus period, we subtracted the mean response from
each individual response. We then Fourier-transformed the mean response
and all individual noise traces to obtain the mean response and noise
spectra. Both membrane potential and spikes are represented in the same
way and therefore were treated identically in our evaluation programs.
This definition of the signal depends on the instantaneous firing rate
carrying all the information in the spike train; therefore, higher
order statistical properties of the spike train such as interspike
intervals are not taken into account. Having determined the ratio of
signal and noise spectra, we then used Equation 8 to estimate an
expected coherence for a purely linear coding scheme given the
signal-to-noise ratio determined experimentally in the way just
described. To calculate the upper and lower bounds of the information
rate, we used Equations 10 and 14, with a value of 50 Hz as the upper
integration limit because this was the highest frequency produced by
our stimulation device.
| |
RESULTS |
Figure 2 summarizes the phenomena
that form the basis for our present investigations. Here, an HS-cell
was stimulated by moving a square-wave grating in the preferred
direction of the cell in front of the ipsilateral eye of the fly. We
used two different velocity profiles: a step function (Fig.
2d) and a pseudorandomly fluctuating function (white noise
velocity) (Fig. 2h). The cellular responses are shown on
top. In all graphs, the cell was visually stimulated. The cell was
additionally manipulated simultaneously with the visual stimulus
through the recording electrode by a positive, depolarizing current
injection of +3 nA (Fig. 2a,e), without any current
injection, i.e., with the cells at resting potential of approximately
50 mV (Fig. 2b,f), and while the cells were
hyperpolarized by injection of 3 nA (Fig. 2c,g).
View larger version (39K):
[in this window]
[in a new window]
|
Figure 2.
Responses of an HS-cell to a step-like
(d) and a pseudorandomly fluctuating velocity profile
(h). The top rows (a, e) show the
responses of the cell to the motion stimuli with an additional current
injection of +3 nA; the third row (c, g) shows the
responses with current injection of 3 nA. b and
f show the response of the cell at rest. Note the small
spike-like events with irregular amplitude in b and
f. By hyperpolarizing the cell these spike-like events turn
into full-blown action potentials (c, g). When the cell is
depolarized, spikes are no longer elicited (a, e).
|
|
At resting potential, HS-cells respond to a step-like preferred
direction motion with a graded shift in membrane potential (Fig.
2b). Superimposed on this graded shift are small spike-like events with an amplitude of ~10-20 mV. These fast and irregular action potentials reflect the existence of voltage-gated ion channels in the membrane of HS-cells. By injecting hyperpolarizing current into
the axon of the HS-cell while simultaneously stimulating the cell by
pattern motion, this so-called mixed response mode can be turned into a
graded response with full-blown action potentials (Fig. 2c).
Under these conditions, active membrane properties obviously contribute
more significantly to the response of the cell. The spikes elicited
under these conditions have an amplitude of >50 mV. In contrast, when
the cell is depolarized by current injection of +3 nA,
voltage-activated sodium channels become inactivated and spikes are no
longer elicited by the motion stimulus (Fig. 2a). When the
step-like pattern is replaced with a pseudorandom velocity profile
displayed in front of the fly, again, full-blown action potentials are
elicited only when the cell is hyperpolarized (Fig. 2g). At
resting or more positive potentials, these action potentials become
smaller and more irregular, and often are hardly discernible from other
membrane fluctuations (Fig. 2e,f).
To assess the coding capability of the cell under these different
response modes, we applied the reverse reconstruction technique (Bialek
et al., 1991; Theunissen et al., 1996) (for details, see Materials and
Methods). Figure 3a-c shows
the impulse responses of the reverse filter obtained under the three
modes of current injection. All filters are non-zero for negative time
values to compensate for the delay of the response with respect to the
stimulus. Furthermore, they all exhibit typical bandpass
characteristics and reveal differences in only small details. The
filter for the depolarized cell has the largest amplitude (Fig.
3a), and for the hyperpolarized cell it has the smallest
amplitude (Fig. 3c). After normalizing for the peak
amplitudes, the following differences in the time course become visible
(Fig. 3d). The filter derived from the cell without current
injection has the shortest half-maximum width, whereas the filter
derived from +3 nA current injection has the broadest. More differences
between the filters can be seen when transformed into Fourier space
(Fig. 3e). The filter for the artificially depolarized cell
amplifies much more than the other filters in the frequency range
between 0.2 and 30 Hz. The reason for that is the reduced amplitude of
the cellular response under these conditions (Fig. 2, compare
e,f). Whereas the amplitude of the filter for the
nonmanipulated cell and the filter for the depolarized cell have about
the same amplitude for frequencies >30 Hz, the filter amplitude of the
hyperpolarized cell already starts to decrease at 10 Hz.
View larger version (32K):
[in this window]
[in a new window]
|
Figure 3.
a-c, Impulse responses of the reverse
filter for the three different experimental conditions. The data are
derived from experiments on 16 HS-cells at rest, 11 hyperpolarized, and
7 depolarized cells. The graphs show the mean values ± SEM. Each
filter has bandpass characteristics. d, Normalized
amplitudes of the impulse responses shown in a-c. e,
Amplitude spectra of the three filters. The filter for the depolarized
cell has the highest amplitude in the low frequency range (dotted
line).
|
|
We calculated the coherence functions between the visual stimulus and
the cellular responses as measured under the different current
conditions (Fig. 4). The coherence values
were highest for HS-cells at resting potential (n = 16 cells) and reached almost 80% for frequencies up to 10 Hz (thick
line). For higher frequencies the coherence fell off rather steeply and
approximated zero level at 50 Hz. When the cells were hyperpolarized
during visual motion stimulation (n = 11), the
coherence values in the lower frequency range were ~15-20% smaller
than when the cells were stimulated at resting potential (thin line).
The coherence values for the cells in the depolarized state
(n = 7) lay in between (dotted line). Thus, motion
information was preserved in the response traces of nonmanipulated
cells with higher accuracy than in the response traces of manipulated
cells, no matter whether the current was a depolarizing or a
hyperpolarizing one. To summarize these points we plotted the mean
coherence value between 0.2 and 10 Hz for the different electrical
manipulations of the cell (Fig. 5). The
mean coherence level clearly is optimal when the cell is at its normal
resting potential of approximately 50 mV. From the coherence
functions we calculated the information rate (lower bound) for the
cells under the various conditions (see Materials and Methods; Eq. 14).
This was done using an upper frequency limit of 50 Hz. For the cells at
rest the lower bound of the information rate was 37 bits/sec and 32 bits/sec for the depolarized cells. The hyperpolarized cells had the
lowest information rate with 20 bits/sec.
View larger version (25K):
[in this window]
[in a new window]
|
Figure 4.
Comparison between the coherence functions under
the three different experimental conditions. Coherence was highest for
HS-cells at rest (thick line; average over n = 16 cells) and reached values of ~0.75 in the low-frequency range.
The root mean square error ( ) between the estimated stimulus
Sest and the stimulus amounted to
79.4 ± 1.1°/sec. For hyperpolarized HS-cells (thin
line, n = 11) the coherence was ~15-20% lower
( = 87.1 ± 0.9°/sec). The values for the depolarized cell
lay in between (dotted line, n = 7). The
root mean square error for the depolarized cell amounted to 82.9 ± 2.2°/sec. The error bars at 1.5 Hz show the SEM for a single
representative frequency.
|
|
View larger version (18K):
[in this window]
[in a new window]
|
Figure 5.
Mean coherence level and SEM between 0.2 and 10 Hz
for the three states of the cell (same data as Fig. 4). The coherence
was highest for HS-cells at rest and reached values of 0.68. For
hyperpolarized HS-cells the mean coherence was 25% lower. The value
for the depolarized cells lay in between.
|
|
Although the coherence reached rather high values in the low frequency
range of ~60% even for the electrically manipulated cells, there
still remained a gap of ~20% as compared with the nonmanipulated
cells. What was the reason for the reduced coherence in cells that were
moved away from resting potential by current injection? The coherence
difference, in principle, could be caused by a decrease in
signal-to-noise ratio and to an increased nonlinearity introduced by
the manipulation of the membrane potential via current injection or
both. To decide which of these sources was the prime reason for the
diminished coherence, the signal (i.e., the mean response) and the
noise spectra in response to repeated stimulus presentations were
measured (Fig. 6a-c). The
noise was calculated as the difference between the individual membrane
response and the average response (see Materials and Methods).
Comparison of the signal and noise spectra for different current
injections revealed that the hyperpolarized cells showed the highest
noise level, whereas the signal was as high as in the cells at rest (Fig. 6, compare a,b). The greater influence of active
membrane processes thus did not enhance the mean response but did
increase the noise level. Depolarized cells exhibited the lowest signal amplitude (Fig. 6c). This might reflect the inactivation of
voltage-dependent channels and the concomitant loss of signal
amplification (Haag and Borst, 1996). In contrast to the hyperpolarized
cells, the noise level was the same as in the nonmanipulated cells. All
of these facts together resulted in the SNRs that are shown in Figure 6d. The SNR was highest for the cells at rest (thick line)
and dropped to the value of 1 at 35 Hz. For the depolarized
(dotted line) and hyperpolarized (thin line) cells, the SNRs were
almost identical. They were significantly lower than for the
nonmanipulated cells and dropped to the value of 1 at frequencies >25
Hz. Thus, electrical manipulation of the membrane potential led to a
reduction of the SNR in both cases, no matter whether the cells were
depolarized or hyperpolarized.
View larger version (33K):
[in this window]
[in a new window]
|
Figure 6.
a-c, Signal and noise spectra for
HS-cells at rest (a, n = 16), when hyperpolarized
(b, n = 11) and when depolarized (c, n = 7). The noise level (dotted line) of depolarized cells was
about the same as the noise level for cells at rest. The noise level of
hyperpolarized cells was increased compared with cells at rest.
d, Signal-to-noise ratios derived from the data shown in
a-c. The cells at rest had a higher signal-to-noise ratio
compared with the electrically manipulated cells.
|
|
All of these results point to a decreased SNR as a possible source of
the reduced coherence. However, as mentioned above, an increased
nonlinearity in the encoding of motion information might yield the same
effect. To assess such possible nonlinearities, we calculated an
expected coherence function from the measured signal and noise spectra
(see Materials and Methods; Eq. 8), assuming a purely linear encoding.
A prerequisite for calculating the expected coherence from the signal
and noise spectra is that the cross-correlation between the stimulus
and the noise is zero. The cross-correlation between the noise and the
stimulus (Fig. 7b) is three
orders of magnitude smaller than the cross-correlation between the
response of the cell and the stimulus (Fig. 7a). To
calculate the upper bound of the information rate with Equation 10 the
noise distribution has to be Gaussian. Figure 7c-e suggests
that the noise is a Gaussian stochastic process and that the injection
of current did not influence the distribution of the noise (Fig.
7c-e). The expected coherence values were highest for
HS-cells at resting potential, reaching values of 90-95% in the
low-frequency range (Fig. 8a).
For the electrically manipulated cells, the respective values were
~10% lower. This point is summarized in Figure
9a where the average expected
coherence between 0.2 and 10 Hz for the electrically manipulated and
the cells at rest is shown. The lower expected coherence of the
electrically manipulated cells reflects the decreased SNR under these
conditions. However, if nothing else had changed in the cells except
for a decreased SNR, the expected coherences should account completely
for the difference between the measured coherence functions of the
cells under the different conditions. To evaluate this question
quantitatively, we calculated the difference between the expected and
the measured coherences of the cells for each condition. This
difference can be regarded as the degree of nonlinearity inherent in
the response of the cell, independent of the actual SNR. If the cell
was perfectly linear, measured and expected coherence functions should
be identical. If there was a nonlinearity in the response and this
nonlinearity did not change by current injection, then the difference
between measured and expected coherence should remain the same,
independent of the experimental condition. Figure 8b shows
the result of our analysis. Here, the difference between measured and
expected coherences are plotted. The nonlinearity was highest in the
low-frequency range for hyperpolarized cells. The depolarized cells and
the cells at rest showed the same amount of nonlinearity. This can also
be seen in the averaged nonlinearity shown in Figure 9b. This finding demonstrates that, in addition to the change of SNR spectra, injection of hyperpolarizing current also led to an increased nonlinearity in the cells. From the expected coherences we calculated the channel capacity (upper bound, Eq. 10) for the cells under the
various conditions, again using an upper frequency limit of 50 Hz. The
channel capacities for the electrically manipulated cells were almost
identical to each other (depolarization: 84 bits/sec;
hyperpolarization: 79 bits/sec) and approximately 30 bits/sec lower
than the rate for the cells at rest (110 bits/sec). Compared with the
lower bound, the values for the upper bound are ~50-80 bits/sec
higher (Fig. 10).
View larger version (27K):
[in this window]
[in a new window]
|
Figure 7.
Cross-correlation between stimulus and mean
response (a) and between stimulus and noise (b).
The values of both curves differ by three orders of magnitude,
indicating that the noise is uncorrelated with the stimulus.
c-e, Probability distribution of the noise for the three
current injections. The solid line shows a Gaussian fit. The
distribution does not change if current is injected.
|
|
View larger version (26K):
[in this window]
[in a new window]
|
Figure 8.
a, Expected coherence functions as
determined from the signal and noise spectra shown in Figure 6. Because
of the highest signal-to-noise ratio for HS-cells at rest the expected
coherence was highest too, for this experimental condition.
b, Nonlinearity as defined by the difference between the
expected and the measured coherence. This nonlinearity is highest when
the cells were permanently hyperpolarized, producing full-blown action
potentials in response to visual stimulation. The error bars at 1.5 Hz
show the SEM for a single representative frequency.
|
|
View larger version (20K):
[in this window]
[in a new window]
|
Figure 9.
a, Mean expected coherence level and
SEM between 0.2 and 10 Hz for the three states of the cell (same data
as Fig. 8a). The coherence was highest for HS-cells at rest
(averaged over n = 16 cells) and reached values of
0.91. The value for depolarized (n = 7) and
hyperpolarized cells (n = 11) were approximately 10%
lower. b, Mean nonlinearity level and SEM between 0.2 and 10 Hz for the three states of the cell (same data as Fig. 8b).
The value for hyperpolarized cells (n = 11) was
highest, whereas the values for depolarized (n = 7) and
cells at rest (n = 16 cells) are almost identical.
(Note different scaling of a and b.)
|
|
View larger version (21K):
[in this window]
[in a new window]
|
Figure 10.
Upper and lower bounds of information rates for
the three states of the cell. The error bars show the SEM.
|
|
To disentangle the information carried by the graded membrane potential
from the information carried by the action potentials that occur in
HS-cells when hyperpolarized, we artificially transformed the analog
membrane signal of HS-cells (Fig.
11b) into a spike train
(Fig. 11c) or a response trace without spikes (Fig.
11a) by applying a threshold operation. Whenever the
membrane potential was above this threshold, the spike was cut out
(Fig. 11a) or a unitary pulse of 1 msec duration and 100 mV
amplitude was added to the output, which was zero otherwise (Fig.
11c). The latter procedure turned the original analog record
of the membrane potential into a binary all-or-nothing signal, the same
way the action potentials of spiking neurons like the H1-cell are
usually recorded extracellularly. The filters (data not shown) and the
coherence for the measured membrane potential and the "spike-less"
potential trace turned out to be identical (Fig.
12). Thus, it seems that the occurrence of spikes is not responsible for the lower coherence of hyperpolarized cells compared with cells at rest.
View larger version (22K):
[in this window]
[in a new window]
|
Figure 11.
Transformation of the graded membrane potential
of a hyperpolarized (3 nA) HS-cell (b) into a response
trace without spikes (a) and a train of action potentials
with unitary amplitude and 1 msec duration (c).
|
|
View larger version (21K):
[in this window]
[in a new window]
|
Figure 12.
Comparison of the coherence functions calculated
for the measured membrane potential of a hyperpolarized HS-cell
(thick line) and the artificially reduced response trace
(thin line).
|
|
To measure how well the information about the stimulus velocity was
retained in this artificial spike train, we applied the same technique
as for the analog response traces. Figure
13a shows the coherence
between stimulus velocity and the analog membrane potential for the
hyperpolarized cell (thin line) and the coherence between stimulus
velocity and the spike train (thick line). In general, the coherence
between the stimulus velocity and the membrane potential was
significantly higher than the coherence between stimulus velocity and
the spike train. An examination of the difference between measured and
expected coherence as a measure of nonlinearity in the signals revealed
that the diminished coherence was only partly attributable to different
degrees of nonlinearity (Fig. 13b). In fact, the degree of
nonlinearity was rather similar under both conditions (note different
scales on the y-axes of Fig. 13a,b). Thus, the
difference between the measured coherences of graded response versus
spike train was largely attributable to a decreased signal-to-noise
ratio after the graded response was thresholded.
View larger version (25K):
[in this window]
[in a new window]
|
Figure 13.
a, Coherence functions calculated for
the membrane potential of hyperpolarized HS-cells (thin
line; n = 6 cells) and spike trains (thick
line; n = 6 cells). The root mean square error for
the membrane potential was 86.2 ± 1.1°/sec and 91.5 ± 1.2°/sec for the spike train. The coherence calculated from the
membrane potential was much higher than the one from the spike train.
b, Nonlinearity, i.e., difference between the expected and
the measured coherence for the membrane potential and the spike train.
The error bars at 1.5 Hz show the SEM for a single representative
frequency.
|
|
In Figure 11b, the spikes of HS-cells can be seen to have
variable amplitudes. To test whether the amplitude of the spikes carries some information, we again transformed the analog membrane potential of hyperpolarized HS-cells into spike trains with unitary duration. In contrast to Figure 11c, the amplitude of the
spikes was not set to a unitary value but left the same as the
amplitude of the original spike. This transformation did not result in
a greater coherence compared with the spike train with unitary
amplitude (data not shown).
Because our previous study (Haag and Borst, 1997) revealed that the
spiking H1-cell exhibited a lower coherence than the graded HS-cell,
the outcome of the comparison between the graded and the thresholded
HS-cell response was not surprising: spikes were elicited only in
response to motion in the preferred direction, whereas the graded
membrane response could be shifted in both directions. Because of the
low spontaneous frequency, there was very little information in the
spike train about motion in the anti-preferred direction. This
limitation of spike trains applied to the spiking H1-cell as well as
the artificially spiking HS-cell. We therefore compared the spikes
obtained by thresholding the response of HS-cells with the spikes as
recorded from the H1-cell. We found that the spiking HS-cell showed a
lower coherence than the H1-cell (Fig.
14a). The main deviation of
the coherences for both cell types was between 1 and 20 Hz. In that
range the measured coherence for the HS-cell was ~20% lower than the
coherence found for the H1-cell. For HS-cells the signal as well as the
noise was much lower than the respective values for the H1-neuron (Fig. 14b). This was attributable to the low average firing rate
found for the HS-cells. H1-cells responded with an average firing rate of 48 ± 1 spikes/sec during the stimulation. HS-cells fired with an average rate of 11 ± 0.5 spikes/sec in response to the
identical stimulus.
View larger version (38K):
[in this window]
[in a new window]
|
Figure 14.
a, Comparison between the coherence
functions calculated from the spike trains of HS-cells (thick
line; n = 6 cells) and H1-cells (thin
line; n = 10 cells). b, Signal and
noise spectra for HS- and H1-cells. The amplitude of the signal and the
noise of HS-cells is much lower compared with H1-cells. c,
Signal-to-noise ratio derived from the data shown in b. d,
Nonlinearity in the responses of HS- and H1-cells.
|
|
When we calculated signal-to-noise ratios, we found that these ratios
were also smaller in HS-cells than in H1-cells (Fig. 14c).
In contrast, the degree of nonlinearity, i.e., the difference between
measured and expected coherence, was about the same for both cell types
(Fig. 14d). From these SNRs, we again calculated channel
capacities for the HS spike traces and H1-cells. The information rate
for HS-cells (upper bound, 67 bits/sec; lower bound, 13 bits/sec) was
lower than for H1-cells (upper bound, 79 bits/sec; lower bound, 23 bits/sec) in the frequency range of 0.25-50 Hz. We also calculated the
amount of information carried by a single action potential simply by
dividing the upper and the lower bound information rate in bits per
second by the number of spikes counted per second. This procedure
resulted in a high information content of approximately 6 bits (upper
bound) and 1.2 bits (lower bound) per spike of the HS-cells as compared
with only 1.7 bits (upper bound) and 0.5 bits (lower bound) per spike
for the H1-cells. Because the original information rates of both cells
were rather similar, the large difference in the information per action
potential was attributable to the low average firing rate found in
HS-cells as compared with H1-cells. To examine the influence of a low
spike rate on the information content, we artificially decreased the
average spike frequency of H1-cells by taking only every fourth spike
of the original record of the motion response into consideration. This reduction of the number of spikes led to an average rate of 12 spikes/sec, which is comparable to the response of HS-cells. The coherence for this artificially reduced spike train of H1-cells together with the coherence for spike trains from H1- and HS-cells is
shown in Figure 15a. The
reduction of the number of spikes mainly led to a decreased coherence
for frequencies between 3 and 30 Hz. The coherence for these higher
frequencies became as low as the coherence of HS-cells (Fig.
15c). Thus, 75% of all the H1 spikes are responsible for
the small improvement of ~20% coherence in this frequency range.
View larger version (23K):
[in this window]
[in a new window]
|
Figure 15.
a, Comparison between the coherence
functions for spike trains of H1- and HS-cells. The reduction of the
number of H1-spikes by 75% (dotted line; n = 10 cells) led to a decreased coherence mainly for frequencies >3 Hz.
It became as low as the coherence of HS-cells (thick line;
n = 6 cells) in this frequency range. b,
Mean coherence level and SEM for H1-, reduced spikes of H1-, and
HS-cells between 0.2 and 3 Hz. c, Mean coherence level and
SEM between 3 and 30 Hz.
|
|
| |
DISCUSSION |
In this paper, we investigated the influence of active membrane
properties on the encoding of stimulus velocity in the neural signals
of a motion-sensitive interneuron of the fly visual system, the
HS-cell. These cells are characterized by a low resting potential (approximately 50 mV) and a graded potential shift with spike-like events superimposed in response to visual motion stimuli. The injection
of depolarizing current while the visual response of the cell was
measured led to a reduction in the number and amplitude of these action
potentials. The injection of hyperpolarizing current induced full-blown
action potentials in response to excitatory visual stimuli. This
allowed us to study the same cell type in different response
regimes.
In this context it is important to ask to what extent the recorded
response properties and resting potentials of HS-cells reflect the
properties of the cell without electrode impalement. Of course,
absolute certainty about this point is impossible, but on the basis of
the following results we are confident that the response properties are
not artificial. (1) Intracellular recordings with the same type of
electrodes from other motion-sensitive interneurons (e.g., H1-cells)
that possess much thinner axons than HS-cells resulted in reliable
resting potentials of approximately 60 mV. Under these conditions,
H1-cells were found to always produce large-amplitude action potentials
(Haag, 1994). (2) Although the action potentials of the spiking H1-cell
can be easily recorded with an extracellular electrode, no such
recordings have ever been reported from HS-cells. (3) The impalement of
the dendrite or the axon of HS-cells by a second intracellular
electrode has no influence on response properties and resting potential
(Haag and Borst, 1996) or the input resistance (J. Haag and A. Borst, unpublished observations) of the cell.
To evaluate the coding quality of the neural signals under the
different conditions caused by the current injection, we calculated the
coherence between the stimulus and responses. This value is confined
between 0 and 1 and is also known as the correlation coefficient in
case of scalar value pairs. It measures the amount of scatter of the
data points around the linear regression line. Clearly, a large scatter
and thus a low coherence can have two causes: (1) there is a large
amount of noise present in the transformation between stimulus and
response, or (2) the transformation from the stimulus to the response
is nonlinear by nature or both. We disentangled these two sources for a
reduced coherence by measuring, in addition to the coherence, the
signal-to-noise ratio in the responses by repeated stimulus
presentations. Assuming a purely linear stimulus-response
relationship, we calculated an expected coherence based on these
measured signal-to-noise ratios. The difference between a value of 1 and the expected coherence thus could be attributed to noise, whereas
the remaining difference between the expected and the measured
coherence had to be caused by nonlinear encoding.
We found that HS-cells without an electrical manipulation represented
motion information with a higher precision than electrically manipulated cells. Depolarization as well as hyperpolarization of the
cells led to a decrease in coherence. In the depolarizing cells, this
decrease was exclusively attributable to a lower signal-to-noise ratio
of the motion responses. Compared with the cells at rest, the noise
level did not change, whereas the amplitude of the mean response
decreased. This was most likely caused by an inactivation of
voltage-dependent channels that normally amplify the motion response of
the cells at rest. Because the noise level was the same for the
depolarized cell and the cell at rest, we conclude that active membrane
processes do not contribute much to the noise level of these cells at
resting potential. In the hyperpolarized cells the decrease of the
coherence was again attributable to a lower SNR, but in addition to an
increased nonlinearity in the response. The lower SNR resulted from an
increased noise level. Thus, the larger contribution of
voltage-dependent processes did not further amplify the mean response
but did enhance the amplitude of the noise. In addition, the response
also became more nonlinear under these conditions. It appears that the
resting potential of the cells represents an ideal set-point where
image velocity can be optimally represented by the membrane potential
of the cell. Thus, in contrast to our expectation, active processes, when tuned in just the right way, do not deteriorate the representation of image velocity in the neural signal of HS-cells by the introduction of nonlinearities or an increase of noise but rather enhance the coding
efficiency by amplifying signals but not noise. This could be
accomplished, e.g., by a regional differentiation between signal and
noise going along with a spatially inhomogeneous distribution of
amplification mechanisms. Whether the dominant noise source is
intrinsic to HS-cells or caused by presynaptic circuitry remains to be
clarified. In any case, it will be important to gain a detailed understanding at the biophysical level of the intricate interplay between the intrinsic active membrane properties of HS-cells and their
synaptic input signals and the consequences for their coding capabilities. Biophysically realistic compartmental models of HS-cells,
which were developed recently in our laboratory (Borst and Haag, 1996;
Haag et al., 1997), shall prove a useful tool in this context.
As a consequence of the maximum SNR of HS-cells at rest, the channel
capacity was also maximum under these conditions. The same was true for
the lower bound on the information rate. When calculating lower and
upper bounds, we restricted the frequency range to an upper limit of 50 Hz, because given a frame rate of 200 Hz of our stimulus device the
highest velocity modulation that we could produce amounted to 50 Hz. We
found a channel capacity at rest of 110 bits/sec. This information rate
is rather low compared with the large monopolar cells of the fly visual
system, which are postsynaptic to the photoreceptors and exhibit,
depending on the mean light level, a temporal low-pass or bandpass
filter characteristic (van Hateren 1992a,b). Large monopolar cells
respond to changes in light intensity by a graded shift of membrane
potential as HS-cells do. The upper limit of the rate at which these
cells transmit information about the light intensity was measured to 1650 bits/sec (de Ruyter van Steveninck and Laughlin, 1996) and thus is
~15 times higher than the maximum possible rate at which HS-cells
could transmit information about the stimulus velocity. This huge
difference probably reflects the fact that in the case of the large
monopolar cells the light intensity is already represented at the
photoreceptor output synapse, whereas the velocity signal is not a
stimulus parameter uniquely encoded by the synaptic input but rather is
being calculated, at least in part, on the dendrite of the LPTCs
(Single et al., 1997).
The comparison between the membrane potential and the artificially
produced spike trains obtained by thresholding the visual responses of hyperpolarized HS-cells revealed that the analog membrane
potential carried more information than the binary all-or-nothing spike
train. This is most likely attributable to the fact that when the
average spontaneous firing rate is low, spikes can transmit information
almost exclusively about one direction of the motion stimulus, whereas
graded responses can follow both directions of the motion stimuli
without any immediate ceiling effects (Haag and Borst, 1997). The
comparison of the spike trains of H1- and HS-cells showed that more
information about the stimulus was retained in the spike train of
H1-cells. This is most likely because of the lower spike frequency of
HS-cells.
Another question arising from this work concerns the way information is
transmitted from HS-cells to postsynaptic cells. If there is a graded
transmitter release (e.g., Angstadt and Calabrese, 1991; Laurent, 1993)
without any threshold, the information contained in the graded membrane
response of HS-cells can be fully transmitted to the next cell. A
threshold operation would introduce further nonlinearities in the
response of the postsynaptic cell to visual motion stimuli. More
importantly, information about the null direction motion that is
inherent only in the graded membrane response of HS-cells but not in
its spike-like depolarization could not be transmitted through this
synapse. This seems to be unlikely, because HS-cells are the only known
horizontal-sensitive large-field output elements of the lobula plate
(Hausen, 1984), and thus flies could not respond to null-direction
motion. However, motion stimuli going from back to front and thus along
the null direction of the HS-cells have been found to elicit
significant optomotor responses in flies (Wehrhahn, 1981; Borst et al.,
1991). The information theoretical analysis of postsynaptic cells will
show how much of the information about null-direction motion that is
inherent in the analog membrane potential of HS-cells is indeed
transmitted to these postsynaptic neurons.
| |
FOOTNOTES |
Received April 2, 1998; revised July 10, 1998; accepted July 14, 1998.
We are grateful to J. P. Miller and F. Theunissen for stimulating
discussions at early stages of this work.
Correspondence should be addressed to Dr. Juergen Haag,
Friedrich-Miescher-Laboratory of the Max-Planck-Society, Spemannstrasse 37-39, D-72076 Tuebingen, Germany.
| |
REFERENCES |
-
Angstadt JD,
Calabrese RL
(1991)
Calcium currents and graded synaptic transmission between heart interneurons of the leech.
J Neurosci
11:746-759[Abstract].
-
Bialek W,
Rieke F
(1992)
Reliability and information transmission in spiking neurons.
Trends Neurosci
15:428-434[Web of Science][Medline].
-
Bialek W,
Rieke F,
de Ruyter van Steveninck R,
Warland D
(1991)
Reading a neural code.
Science
252:1854-1857[Abstract/Free Full Text].
-
Borst A,
Egelhaaf M
(1989)
Principles of visual motion detection.
Trends Neurosci
12:297-306[Web of Science][Medline].
-
Borst A,
Egelhaaf M
(1990)
Direction selectivity of fly motion-sensitive neurons is computed in a two-stage process.
Proc Natl Acad Sci USA
87:9363-9367[Abstract/Free Full Text].
-
Borst A,
Haag J
(1996)
The intrinsic electrophysiological characteristics of fly lobula plate tangential cells. I. Passive membrane properties.
J Comput Neurosci
3:313-336[Web of Science][Medline].
-
Borst A,
Buchstäber F,
Egelhaaf M
(1991)
Spatial integration of visual motion information in the optomotor response of the housefly.
In: Synapse
 TransmissionModulation (Elsner N, Penzlin H, eds, p. 262. Stuttgart: Thieme. -
Bullock TH
(1981)
Spikeless neurones: where do we go from here?
In: Neurones without impulses (Roberts A,
Bush BMH,
eds), pp 269-284. Cambridge, UK: Cambridge UP.
-
Callaway JC,
Ross WN
(1995)
Frequency-dependent propagation of sodium action potentials in dendrites of hippocampal CA1 pyramidal neurons.
J Neurophysiol
74:1395-1403[Abstract/Free Full Text].
-
Cover TM,
Thomas JA
(1991)
In: Elements of information theory. New York: Wiley.
-
de Ruyter van Steveninck RR,
Laughlin SB
(1996)
The rate of information transfer at graded-potential synapses.
Nature
379:642-645.
-
Egelhaaf M,
Borst A,
Reichardt W
(1989)
Computational structure of a biological motion detection-detection system as revealed by local detector analysis in the fly's nervous system.
J Opt Soc Am A
6:1070-1087[Web of Science][Medline].
-
Gabbiani F,
Metzner W,
Wessel R,
Koch C
(1996)
From stimulus encoding to feature extraction in weakly electric fish.
Nature
384:564-567[Medline].
-
Haag J (1994) Aktive und passive Membraneigenschaften
bewegungsempfindlicher Interneurone der Schmeißfliege Calliphora
erythrocephala. PhD thesis, Tuebingen.
-
Haag J,
Borst A
(1996)
Amplification of high-frequency synaptic inputs by active dendritic membrane processes.
Nature
379:639-641.
-
Haag J,
Borst A
(1997)
Encoding of visual motion information and reliability in spiking and graded potential neurons.
J Neurosci
17:4809-4819[Abstract/Free Full Text].
-
Haag J,
Theunissen F,
Borst A
(1997)
The intrinsic electrophysiological characteristics of fly lobula plate tangential cells. II. Active membrane properties.
J Comput Neurosci
4:349-369[Web of Science][Medline].
-
Hausen K
(1982a)
Motion sensitive interneurons in the optomotor system of the fly. I. The horizontal cells: structure and signals.
Biol Cybern
45:143-156[Web of Science].
-
Hausen K
(1982b)
Motion sensitive interneurons in the optomotor system of the fly. II. The horizontal cells: receptive field organization and response characteristics.
Biol Cybern
46:67-79[Web of Science].
-
Hausen K
(1984)
The lobula-complex of the fly: structure, function and significance in visual behaviour.
In: Photoreception and vision in invertebrates (Ali MA,
ed), pp 523-559. New York: Plenum.
-
Hengstenberg R
(1977)
Spike response of "non-spiking" visual interneurone.
Nature
270:338-340[Medline].
-
Hirsch JA,
Gilbert CD
(1991)
Synaptic physiology of horizontal connections in the cat's visual cortex.
J Neurosci
11:1800-1809[Abstract].
-
Laurent G
(1993)
A dendritic gain control mechanism in axonless neurons of the locust, Schistocerca Americana.
J Physiol (Lond)
470:45-54[Abstract/Free Full Text].
-
Press WH,
Flannery BP,
Teukolsky SA,
Vetterling WT
(1988)
In: Numerical recipes in C. The art of scientific computing. Cambridge, UK: Cambridge, UP.
-
Shannon CE,
Weaver W
(1949)
In: The mathematical theory of communication. Chicago: The University of Illinois.
-
Rieke F,
Warland D,
de Ruyter van Steveninck R,
Bialek W
(1997)
In: Spikes: exploring the neural code. Cambridge, MA: MIT.
-
Single S,
Haag J,
Borst A
(1997)
Dendritic computation of direction selectivity and gain control in visual interneurons.
J Neurosci
17:6023-6030[Abstract/Free Full Text].
-
Spruston N,
Schiller Y,
Stuart G,
Sakmann B
(1995)
Activity-dependent action potential invasion and calcium influx into hippocampal CA1 dendrites.
Science
268:297-300[Abstract/Free Full Text].
-
Stuart GJ,
Sakmann B
(1994)
Active propagation of somatic action potentials into neocortical pyramidal cell dendrites.
Nature
367:69-72[Medline].
-
Theunissen F (1993) An investigation of sensory coding
principles using advanced statistical techniques. PhD dissertation,
University of California at Berkeley.
-
Theunissen F,
Roddey JC,
Stufflebeam S,
Clague H,
Miller JP
(1996)
Information theoretic analysis of dynamical encoding by four primary sensory interneurons in the cricket cercal system.
J Neurophysiol
75:1345-1364[Abstract/Free Full Text].
-
van Hateren JH
(1992a)
Theoretical predictions of spatiotemporal receptive fields of fly LMCs, and experimental validation.
J Comp Physiol [A]
171:157-170.
-
van Hateren JH
(1992b)
Real and optimal neural images in early vision.
Nature
360:68-70[Medline].
-
Wehrhahn C
(1981)
Fast and slow flight torque responses in flies and their possible role in visual orientation.
Biol Cybern
40:213-221.
-
Yuste R,
Tank DW
(1996)
Dendritic integration in mammalian neurons, a century after Cajal.
Neuron
16:701-716[Web of Science][Medline].
-
Yuste R,
Gutnick MJ,
Saar D,
Delaney KR,
Tank DW
(1994)
Ca2+ accumulations in dendrites of neocortical pyramidal neurons: an apical band and evidence for two functional compartments.
Neuron
13:23-43[Web of Science][Medline].
Copyright © 1998 Society for Neuroscience 0270-6474/98/18197972-15$05.00/0
This article has been cited by other articles:
|
|
|
|
 
K. Nordstrom and D. C O'Carroll
The motion after-effect: local and global contributions to contrast sensitivity
Proc R Soc B,
May 7, 2009;
276(1662):
1545 - 1554.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
J. Haag and A. Borst
Electrical Coupling of Lobula Plate Tangential Cells to a Heterolateral Motion-Sensitive Neuron in the Fly
J. Neurosci.,
December 31, 2008;
28(53):
14435 - 14442.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
J. Grewe, N. Matos, M. Egelhaaf, and A.-K. Warzecha
Implications of Functionally Different Synaptic Inputs for Neuronal Gain and Computational Properties of Fly Visual Interneurons
J Neurophysiol,
October 1, 2006;
96(4):
1838 - 1847.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
K. Karmeier, J. H. van Hateren, R. Kern, and M. Egelhaaf
Encoding of Naturalistic Optic Flow by a Population of Blowfly Motion-Sensitive Neurons
J Neurophysiol,
September 1, 2006;
96(3):
1602 - 1614.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
M. A. Freed
Quantal Encoding of Information in a Retinal Ganglion Cell
J Neurophysiol,
August 1, 2005;
94(2):
1048 - 1056.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
J. H. van Hateren, R. Kern, G. Schwerdtfeger, and M. Egelhaaf
Function and Coding in the Blowfly H1 Neuron during Naturalistic Optic Flow
J. Neurosci.,
April 27, 2005;
25(17):
4343 - 4352.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
R. A. DiCaprio
Information Transfer Rate of Nonspiking Afferent Neurons in the Crab
J Neurophysiol,
July 1, 2004;
92(1):
302 - 310.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
N. K. Dhingra and R. G. Smith
Spike Generator Limits Efficiency of Information Transfer in a Retinal Ganglion Cell
J. Neurosci.,
March 24, 2004;
24(12):
2914 - 2922.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
C. L. Passaglia and J. B. Troy
Information Transmission Rates of Cat Retinal Ganglion Cells
J Neurophysiol,
March 1, 2004;
91(3):
1217 - 1229.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
L. Paninski, M. R. Fellows, N. G. Hatsopoulos, and J. P. Donoghue
Spatiotemporal Tuning of Motor Cortical Neurons for Hand Position and Velocity
J Neurophysiol,
January 1, 2004;
91(1):
515 - 532.
[Abstract]
[Full Text]
|
|
|
|
|
|
|
J. H. van Hateren, L. Ruttiger, H. Sun, and B. B. Lee
Processing of Natural Temporal Stimuli by Macaque Retinal Ganglion Cells
J. Neurosci.,
November 15, 2002;
22(22):
9945 - 9960.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
J. Haag and A. Borst
Recurrent Network Interactions Underlying Flow-Field Selectivity of Visual Interneurons
J. Neurosci.,
August 1, 2001;
21(15):
5685 - 5692.
[Abstract]
[Full Text]
[PDF]
|
|
|
|
|
|
|
F Gabbiani and W Metzner
Encoding and processing of sensory information in neuronal spike trains
J. Exp. Biol.,
January 5, 1999;
202(10):
1267 - 1279.
[Abstract]
[PDF]
|
|
|
|
|
Top
Abstract
Introduction
![&ggr;<SUB><UP>exp</UP></SUB><SUP>2</SUP>(f)=<FR><NU>⟨S<SUB><UP>i</UP></SUB>*(f)·[G<SUB><UP>fwd</UP></SUB>(f)·S<SUB><UP>i</UP></SUB>(f)+N<SUB><UP>i</UP></SUB>(f)]⟩</NU><DE>⟨S<SUB><UP>i</UP></SUB>*(f)·S<SUB><UP>i</UP></SUB>(f)⟩</DE></FR>.](/content/vol18/issue19/fulltext/7972/img006.gif)
![<FR><NU>⟨[G<SUB><UP>fwd</UP></SUB>*(f)·S<SUB><UP>i</UP></SUB>*(f)+N<SUB><UP>i</UP></SUB>*(f)]·S<SUB><UP>i</UP></SUB>(f)⟩</NU><DE>⟨[G<SUB><UP>fwd</UP></SUB>(f)·S<SUB><UP>i</UP></SUB>*(f)+N<SUB><UP>i</UP></SUB>*(f)]·[G<SUB><UP>fwd</UP></SUB>(f)·S<SUB><UP>i</UP></SUB>(f)+N<SUB><UP>i</UP></SUB>(f)]⟩</DE></FR>.](/content/vol18/issue19/fulltext/7972/img007.gif)
