Download Noise and Coupling Affect Signal Detection and Bursting in a

J Neurophysiol
88: 2598 –2611, 2002; 10.1152/jn.00223.2002.
Noise and Coupling Affect Signal Detection and Bursting in a
Simulated Physiological Neural Network
WILLIAM C. STACEY1,2 AND DOMINIQUE M. DURAND1
Department of Biomedical Engineering, Case Western Reserve University; and 2Department of Neurology, University
Hospitals of Cleveland, Cleveland, Ohio 44106
1
Received 27 March 2002; accepted in final form 9 July 2002
influence signal detection in this system (Gluckman et al. 1996,
1998; Stacey and Durand 2000) using endogenous noise
sources (Stacey and Durand 2001). Even subtle changes above
resting noise variance can improve the ability of neurons to
detect subthreshold signals. The influence of noise on signal
detection in CA1 is therefore quite significant and relevant to
information processing in the brain.
The SR relation between signal-to-noise ratio (SNR) and
noise for a neuron is given in Eq. 1. The SNR is 0 without
added noise, rises quickly to a maximum peak with the addition of noise, and gradually decreases to 1 (D ⫽ noise intensity,
⑀ ⫽ signal strength, and ⌬U ⫽ threshold barrier height) as
noise overcomes the output
冉 冊
⑀ ⌬U ⫺⌬U/D
e
D
2
SNR⬀
(1)
The effect of noise on detection of subthreshold signals in
threshold-detecting systems, or stochastic resonance (SR), has
been investigated in several physiological and nonphysiological systems. SR theory was originally developed to describe a
single bistable element (Benzi et al. 1981; Dykman et al. 1995;
Moss et al. 1994) and was later expanded to describe a monostable system like a neuron (Stocks et al. 1993; Wiesenfeld et
al. 1994). Both computer simulations and in vitro experiments
have been used to demonstrate SR behavior in a number of
neural systems (Braun et al. 1994; Bulsara et al. 1991; Collins
et al. 1996; Douglass et al. 1993; Levin and Miller 1996; Pei et
al. 1996; Russell et al. 1999). Recent studies in the CA1 layer
of the hippocampus have indicated that SR can function to
Examples of this curve are presented in several of the later
figures in this paper. SR has been found to have novel properties on larger numbers of elements. SR in an array of elements with a summed output generates an increased SNR to a
broader range of noise (Chialvo et al. 1997; Collins et al.
1995), although this simulated finding was not found experimentally in cochlear nerves (Morse and Roper 2000). When
the elements are coupled, as in array-enhanced SR (AESR), the
signal detection improvement can vary depending on how the
coupling is performed (Inchiosa and Bulsara 1995a,b; Kanamaru et al. 2001; Lindner et al. 1995; Locher et al. 1996). In
this paper, we have chosen a coupling source to simulate
electrical connections between closely neighboring cells (see
METHODS). AESR has important implications in the nervous
system, because it can make a neural network sensitive to a
broad range of input signals when noise is present (Stocks and
Mannella 2000).
Noise alone can produce oscillatory behavior in a single
element under certain conditions, an effect closely related to
SR (Gang et al. 1993; Lee et al. 1998). Another similar effect
occurs in coupled arrays of neural models that are not presented with any periodic signal: they are able to respond with
synchronous, near-periodic outputs when noise is added (Hu
and Zhou 2000; Pham et al. 1998; Pradines et al. 1999; Wang
et al. 2000). These effects have been called coherence resonance (CR) or autonomous SR. Throughout this work, CR is
Address for reprint requests: D. M. Durand, Dept. of Biomedical Engineering,
Neural Engineering Center, Case Western Reserve University, CB Bolton Rm.
3510, 10900 Euclid Ave., Cleveland, OH 44106 (E-mail: [email protected]).
The costs of publication of this article were defrayed in part by the payment
of page charges. The article must therefore be hereby marked ‘‘advertisement’’
in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
INTRODUCTION
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0022-3077/02 $5.00 Copyright © 2002 The American Physiological Society
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Stacey, William C. and Dominique M. Durand. Noise and coupling
affect signal detection and bursting in a simulated physiological neural
network. J Neurophysiol 88: 2598 –2611, 2002; 10.1152/jn.00223.2002.
Signal detection in the CNS relies on a complex interaction between
the numerous synaptic inputs to the detecting cells. Two effects,
stochastic resonance (SR) and coherence resonance (CR) have been
shown to affect signal detection in arrays of basic neuronal models.
Here, an array of simulated hippocampal CA1 neurons was used to
test the hypothesis that physiological noise and electrical coupling can
interact to modulate signal detection in the CA1 region of the hippocampus. The array was tested using varying levels of coupling and
noise with different input signals. Detection of a subthreshold signal
in the network improved as the number of detecting cells increased
and as coupling was increased as predicted by previous studies in SR;
however, the response depended greatly on the noise characteristics
present and varied from SR predictions at times. Careful evaluation of
noise characteristics may be necessary to form conclusions about the
role of SR in complex systems such as physiological neurons. The
coupled array fired synchronous, periodic bursts when presented with
noise alone. The synchrony of this firing changed as a function of
noise and coupling as predicted by CR. The firing was very similar to
certain models of epileptiform activity, leading to a discussion of CR
as a possible simple model of epilepsy. A single neuron was unable to
recruit its neighbors to a periodic signal unless the signal was very
close to the synchronous bursting frequency. These findings, when
viewed in comparison with physiological parameters in the hippocampus, suggest that both SR and CR can have significant effects on
signal processing in vivo.
NOISE AND COUPLING EFFECTS IN A NEURAL NETWORK
METHODS
Synaptic events generating the periodic input signal and random
events were simulated using AMPA synapse functions (Destexhe et
al. 1998). The amplitude of synaptic events was modulated by changing the maximum conductance (gmax) of the AMPA current. The
periodic signal input was positioned on the distal branch of the apical
dendrite of each cell and the amplitude adjusted to a subthreshold
level (4 nS). The stimulus was a pulse train (Stacey and Durand 2000)
of 2.5 Hz except where otherwise noted. At 2.5 Hz, the neurons were
able to return to baseline between pulses to avoid any frequencydependent effects of the signal. The signal was applied either to all
neurons simultaneously (“global input”) or to one neuron alone (“single input”; Fig. 1B). Noise was added to simulate any synaptic activity
that is uncorrelated to the periodic signal. The noise was generated by
55 presynaptic axons, 10 of which contacted between three and nine
different CA1 cells as en passant axons (e.g., Fig. 1C), while the
remainder made independent connections. The en passant connections
were generated randomly and did not cause any significant correlation
among the CA1 cells. This connection scheme simulates the connectivity that exists in a rat hippocampal slice (Andersen 1990), where
neighboring CA1 cells often receive the same presynaptic signal
(Bernard and Wheal 1994).
Noise
Simulations were performed for two values of maximum conductance (gmax) at the noise synapses: 0.22 and 1 nS. These two values
encompass the experimental range of miniature excitatory postsynaptic potentials (minis; gmax ⫽ 0.21–1 nS) (Bekkers and Stevens 1989;
Destexhe et al. 1998; McBain and Dingledine 1992; Stricker et al.
1996). Quantal size of random events were Poisson distributed
(mean ⫽ 0.3). Noise variance (D, or noise intensity, in Eq. 1) was
modulated by changing the mean firing frequency of the noise synapses. The firing of each presynaptic axon, which evoked a synaptic
event, was calculated independently by comparing a threshold with a
uniform random number generated at each time step (Stacey and
Durand 2000). Changing this threshold thus changed the probability
of firing at each time step. The characteristics of the synaptic noise
current are shown in Fig. 2, which spans the entire range of noise
simulated. Noise bandwidth and spectral power increased as expected
with increased firing probability. Increasing D increased the mean
depolarizing current in a nearly linear fashion. The noise source had
some low-pass filtering effects due to the frequency characteristics of
the AMPA synapse, but they were present for much higher levels of
D than those used in these simulations. Also of note is the different
response to noise with gmax ⫽ 0.22 nS versus 1 nS (see DISCUSSION).
Noise was locally uncorrelated (Lindner et al. 1995) except the minor
effect from synapses connected by en passant axons. There was no
time delay between the multiple synapses contacted by the en passant
axons. Synaptic noise variance was computed from recordings of the
synaptic current.
Coupling
Coupling between each pair of cells was modeled as an additional
voltage controlled current source (I icoupling) in the soma of each cell
(Eq. 2). Current effects from all neighboring neurons were summed to
produce a single current input at the soma
Network simulation
冘
N⫺1
An array of 10 simulated CA1 neurons previously described (Stacey and Durand 2000, 2001) was implemented using NEURON
software (Hines 1993). The neurons (Fig. 1A) had membrane parameters derived from experimental data (Table 1) and were identical to
each other. Each neuron contained an active soma with one sodium,
one calcium, and four active potassium channels (Warman et al. 1994)
as well as passive dendrites. Voltage data from the soma and current at
the synapses were recorded separately with a sampling rate of 2,000 Hz.
J Neurophysiol • VOL
i
I coupling
⫽
a 䡠 cij共Vj ⫺ Erest兲
共j ⫽ i兲
(2)
j⫽0
f C共cij兲 ⫽
1
Cmax ⫺ 0
all cij ⫽ cji
(3)
The current amplitude was determined by the driving potential of the
coupled neuron (Vj ⫺ Erest) and a coupling weight (cij). The coupling
weight, cij, was bidirectional and randomly assigned from the uniform
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used to describe only the case of a coupled network of neurons
and autonomous SR that of a single neuron. In CR, the ability
of noise to produce a synchronous, periodic response in the
system is dependent on the noise characteristics and coupling
between the neurons. When analyzed in the frequency domain,
a sharp peak is produced as the system becomes more periodic,
which can be used as a measure of synchrony. The synchrony
follows a bell-shaped curve as noise intensity increases
(Neiman et al. 1997).
While SR in a single element has been verified experimentally using several different models, experimental evidence
demonstrating the presence or utility of AESR or CR in physiological neural systems is lacking. Although several different
mathematical neural models have been used to investigate
AESR and CR, even simultaneously (Lindner and Schimansky-Geier 1999, 2000), there has been little correlation of the
effects with physiological parameters such as membrane dynamics, channels, and voltage. Rationalization and evaluation
of the simulated data in light of its physiological ramifications
has been difficult without these analyses or experimental verification. It is a major goal of this work to provide such
correlations to physiological data by using a neural model that
has been verified experimentally and that utilizes cellular parameters.
The beneficial effect of noise on signal detection has intriguing implications in the brain. Neurons such as CA1 pyramidal
cells in the hippocampus are responsible for integration of
signals from many different sources (Andersen 1990). Signal
detection in these cells is complicated by the presence of noise
(Bekkers et al. 1990; Destexhe and Paré 1999; Kamondi et al.
1998; Sayer et al. 1989;), the large number of synapses, and
dendritic attenuation (Spruston et al. 1993), a situation that
makes them ideal candidates for physiological SR (Stacey and
Durand 2000, 2001). It is quite possible that AESR and CR
also have physiological effects in the CA1 layer due to the
network organization. The presence of these phenomena in the
brain would have broad implications for normal signal detection as well as for abnormal detection such as epileptogenesis.
In this paper, we seek to analyze the effects of noise and
coupling on signal detection in an array of simulated CA1
cells. This simulation is based on a detailed implementation of
specific cellular parameters in CA1 cells (Warman et al. 1994)
that have been verified experimentally (Stacey and Durand
2000, 2001). Noise and coupling have been added according to
physiological conditions and have been carefully verified with
experimental data. The model is used to test the hypothesis that
noise and coupling can work synergistically to improve detection of a signal in the CA1 layer of the rat hippocampus
through the effects of AESR and CR.
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W. C. STACEY AND D. M. DURAND
interval (0 3 Cmax), as shown by the probability density function in
Eq. 3. In other words, each cell “i” received a coupling current from
each of the nine cells “j”, and the weight of coupling current was
randomized on a scale determined by the value of Cmax. The coupling
strength was multiplied by the arbitrary constant “a” to allow simple
implementation in NEURON. The coupled current caused the potential to move in the same direction as the driving potential. This method
produced an instantaneous change in somatic current that was dependent on the voltage change of another cell, which is similar to
TABLE
1. CA1 model specifications
Dendrite
Rm, ⍀cm2
Cm, ␮F/cm2
Soma
Rm, ⍀cm2
Cm, ␮F/cm2
Ri, ⍀cm
Dimensions
Soma
Basal
1° Apical
Auxiliary 2° apical
Main 2° apical
14,000
2
28,000
1
150
Diam
(␮m)
L (␮m)
Segments
␭
15
4
3.5
3.13
1
20
250
400
400
400
1
5
10
5
5
0.011
0.259
0.443
0.468
0.828
Parameters for the CA1 model. The model architecture is identical to that in
(Stacey and Durand 2000). The implementation differed in that noise variance
was modulated by changing the mean frequency of firing.
J Neurophysiol • VOL
physiological forms of electrical coupling (see DISCUSSION). The distribution did not take into account any spatial effects suggested by the
linear schematization in Fig. 1, B and C. The coupling mechanism did
not generate any shunting between the cells that would diminish the
signal strength (Lindner et al. 1995).
Signal analysis
The membrane voltages from all neurons were summed to produce
a network voltage response: Vsum ⫽ Vi (Collins et al. 1995). These
n
data were mean-detrended (Vsum(rest) ⫽ 0 mV) to remove the resting
voltage contribution from each cell. Any subthreshold voltages (Vsum
ⱕ 70 mV) were then masked to a value of 0 while preserving the
amplitude of suprathreshold responses. In this manner, the only nonzero data contained in the output was from action potentials and
preserved their net amplitude above resting voltage to determine when
multiple cells were firing. The resulting output time series was used to
obtain the power spectrum (Wiesenfeld and Moss 1995) using the
PSD function in Matlab (50,000-point Hanning window; 45,000-point
overlap). The power at the input signal frequency (2.5 Hz) was
divided by the baseband power (noise) in the region near 2.5 Hz to
obtain the SNR. All data in this paper are averages of three independent simulations. The SNR obtained from each value of noise variance
was plotted and used as data for fitting to the SR equation (Stacey and
Durand 2000). In multiple-trace plots the SD is omitted from some
traces for clarity, but in all cases the omitted deviations were comparable to those included in the figure and quite small. The values for ⑀
and ⌬U (Eq. 1) obtained from a least-square error fit were used to
compare the SR responses.
For calculation of coherence, the output of a network presented
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FIG. 1. CA1 model and network organization. A: schematic of an individual CA1 cell, showing positions of noise
sources (stars) and subthreshold signal (S). Parameters of the
neural segments are in Table 1. B: linear representation of the
network, showing how the global signal source axon contacted all 10 cells, and the single source only contacted the
cell at the far left, the “signal cell.” The network was implemented as a randomly arranged cluster: the spatial organization implied by the schematization was not included in the
model. C: implementation of the network. Diagram shows 2
of the en passant noise axons that contacted multiple cells.
Membrane voltage of each cell was summed to produce the
output of the network. D: calculation of synchrony. Power
spectrum of the system output produced frequency spikes
when coherence resonance (CR) was present. Synchrony (␤)
was evaluated for each simulation independently (Eq. 4).
NOISE AND COUPLING EFFECTS IN A NEURAL NETWORK
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FIG. 2. Noise characteristics. A: mean was calculated for the current at 1 AMPA noise synapse for the
range of noise variances used in the simulations. The
noise source behaved differently to the 2 noise signals due to characteristics of the AMPA synapse.
Higher levels of variance were required for the 1-nS
simulations (see RESULTS). B: power spectrum: data
for 0.22 nS noise source with 2 mean firing frequencies. Bandwidth and spectral power increased for the
400-Hz case. Raw data: current from the synapse for
each frequency. Trace is 0.5 s.
␤ ⫽ h 䡠 ␻p/⌬␻
(4)
RESULTS
The array of neurons was configured in four basic forms.
First, the number of neurons in the network was varied without
any coupling as the subthreshold signal and noise were presented to all neurons simultaneously. Second, with 10 neurons,
coupling was varied as all neurons received the signal and
noise again. These two configurations were designed to test the
effect of varying coupling and the number of neurons on
detection of a global signal. Third, the second simulation was
repeated with signal removed from nine of the neurons. This
configuration tested the ability of noise and coupling to affect
recruitment of a network to a signal present at only one cell.
Results were first obtained for these three configurations using
noise with gmax ⫽ 0.22 nS and then repeated for 1 nS. Fourth,
the signal was removed entirely and the 10-neuron network
was presented with varying degrees of coupling and noise
(0.22-nS source). This configuration was used to investigate
oscillatory behavior that was present in many of the previous
simulations.
Effect of multiple uncoupled neurons on subthreshold signal
detection
The network was first used to test the effect of the number
of detecting neurons on detection of subthreshold signals.
Simulations with 1, 3, 5, and 10 neurons were configured with
global inputs without coupling. Noise with mini-amplitude of
0.22 nS was applied. Several levels of noise variance were
added to the system along with the subthreshold signal. The
response shows that noise can improve the network’s detection
of the subthreshold signal (Fig. 3A). As the number of neurons
J Neurophysiol • VOL
increased, the amplitude of the response increased and detection of the input signal was improved (Fig. 3B). Increasing
noise variance produced the typical SR curve and peak SNR
increased as more cells were added (Fig. 3C). Addition of noise
to the system produced peak SNR values of 174, 245, and 467
for 1, 3, and 10 cells, respectively. SNR for 5 cells were
between those for 3 and 10 cells at all noise levels (data not
shown). For variance ⬎15 pA2, the SNR was below the level
predicted by SR. This was due to repetitive firing (oscillations)
of the network present at high noise variance, which increased
the background noise and will be discussed later. The center of
the SR peak did not change with the number of neurons, but the
amplitude varied with N. The change corresponded to an
increasing value for the signal strength (⑀) in the SR equation
(Eq. 1). These results show that arrays of uncoupled neurons,
even as small as three, have an increased ability to improve
detection of a global, subthreshold signal for low noise levels,
but detection can be poorer than predicted by SR theory at high
noise variance for noise with quantal amplitude on the low end
of physiological range.
Effect of coupling on signal detection
The network was then used to test the effects of electrical
coupling between cells when all neurons received the subthreshold input. Coupling was added and modulated between
cells in the 10-neuron array with the same noise source as
before. The network was tested with the global input configuration with four levels of Cmax: 0, 0.1, 0.3, and 0.5. These
levels were chosen to span a spectrum of physiological coupling. The maximum coupling (cij ⫽ 0.5) produced a 0.5-mV
depolarization due to an action potential in a coupled cell,
which was well within the physiological range (see DISCUSSION).
The effect of different coupling levels on the network’s detection of a subthreshold signal in the presence of noise is shown
in Fig. 4A. For noise (10.2 pA2) that produced peak SNR, in the
uncoupled network the signal was detected but the latency
between the first and last cell to fire was often ⬎20 ms (first
trace). In the coupled network, there was a more coherent
response: all 10 neurons tended to fire within a 10-ms span
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with noise but no periodic signal was evaluated. The power spectrum
of each simulation was computed as described above and then analyzed individually. Noise-induced synchrony appeared as a spike in
the power spectrum. Synchrony from this resonant peak was measured
using the coherence factor ␤ in Eq. 4, where h is the height of the
peak, ␻p is the frequency of the peak, and ⌬␻ is the width at half-peak
height (Gang et al. 1993)
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W. C. STACEY AND D. M. DURAND
coupling (SNR ⫽ 7.3 for Cmax ⫽ 0.5 and 102 with Cmax ⫽ 0)
because the signal was overcome by the coherent oscillations.
These results indicate that coupling can greatly affect signal
detection and SR in the network. As the coupling rises, the
network becomes increasingly more sensitive to noise— both
to improve detection with low noise variance and hamper
detection at higher variance.
Effect of noise and coupling on signal detection by a single
neuron in a network
SIGNAL DETECTION OF NETWORK WITH LOW NOISE VARIANCE.
FIG. 3. Effect of number of cells in the network. A: top: raw data showing
the trace of a single cell with a subthreshold signal. Bottom: input signal. When
noise was applied, the signal was detected. B: raw data. As in A, there was no
response before the addition of noise (data not shown). As the number of cells
increased, there was a concurrent increase in both the number of action
potentials in phase with the signal and random action potentials. However, the
increased power in phase with the signal was stronger, leading to an improved
signal-to-noise ratio (SNR). Note that the output was the sum of the array, so
multiple concurrent action potentials appear as spikes ⬎100 mV. Trace is for
5 s of data. C: effect of number of cells on signal detection. Data are fit to the
stochastic resonance (SR) equation. Increasing the number of detecting elements changed the peak without changing the resonant center. All data strayed
below SR theory at high noise. Bars indicate 1 SD for the 10-cell data.
Comparable bars on the other data were removed for clarity.
(second trace). With high levels of noise (17 pA2), extraneous
action potentials were generated, an effect worsened by increased coupling (third and fourth traces). Thus small levels of
coupling and noise improved the signal detection, but as both
increased the network fired clustered action potentials that
overwhelmed the input signal. Results from 0.1 and 0.3 coupling levels (not shown) had a similar relationship, with values
lying between the data shown for Cmax ⫽ 0 and 0.5.
The SNR plots show the effect of coupling on signal detection (Fig. 4B). Coupling significantly increased the network
sensitivity to noise, as shown by an increased SNR at lower
noise levels and a sharper decline in SNR after peak detection.
The peak SNR for maximum coupling (Cmax ⫽ 0.5) was
significantly higher than that with zero coupling (1,208 vs.
467) and occurred at lower noise (5.1 vs. 13.6 pA2). The
coupling therefore caused a shift in the SR curve up and to the
left (decreased ⌬U, increased ⑀). However, with higher noise
variance (17 pA2), the SNR was greatly decreased for higher
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The network response to the single input at low noise levels is
shown in Fig. 5A. For low-to-moderate coupling (Cmax ⱕ 0.9),
the network did not synchronize to the signal. The signal cell
was the only cell able to detect the input at low noise levels.
Increased coupling improved detection at the signal cell (Fig.
5A, first and second traces). The SNR plots (Fig. 5B) show that
increased coupling shifted the SR curves up and to the left,
corresponding to improved detection at these low noise levels.
For comparison, the response of a lone cell to the same noise
levels is included. These results show that even small coupling
levels are able to improve a cell’s ability to detect a subthreshold signal in a noisy system through SR.
SIGNAL DETECTION OF NETWORK WITH HIGH NOISE VARIANCE.
Whereas the network’s improved sensitivity to noise was beneficial for low noise, it greatly diminished SNR at higher
levels. Figure 5, C and D, shows that increased coupling
produced oscillations that corrupt signal detection and depart
from the SR equation. For all levels of coupling tested, the
SNR was once more below the predicted SR value at high
noise. An additional effect occurred independent of noise oscillations: the added noise from the nine other cells also decreased SNR. For this reason, detection with zero coupling was
worse than for a single cell as noise increased. The driving cell
is therefore able to use coupling to enhance its own signal
detection at low noise, but is normally unable to overcome the
additional noise and the oscillations to recruit the other cells.
Effect of noise and coupling on recruitment in the network
The same configuration was also examined for the ability of
the single cell to recruit its neighbors to the subthreshold
signal. Recruitment was defined as more than one cell firing in
response to the input signal and producing an increased SNR.
Under most circumstances, there was no significant recruitment in this network, as seen by the action potentials firing
independently and the low SNR values in Fig. 5. As in the
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The simulation was then performed with the single input
configuration (Fig. 1B) to test the hypothesis that a single cell
can detect a subthreshold signal and recruit its neighboring
cells through the combined effects of SR and coupling. Values
of Cmax ⫽ 0, 0.1, 0.3, 0.5, 0.7, 0.9, 2.5, and 3 were chosen to
encompass all levels of physiological coupling in the system:
from 0 coupling to 1:1 coupling of action potentials for cij ⬎
2.5. The maximum depolarization without transmitting an action potential was 1.8 mV (cij ⫽ 2.5). Note that, due to
randomization, the values of cij for Cmax ⫽ 3 were usually
⬍2.5. This situation is representative of a single neuron receiving a signal surrounded by several quiescent neurons with
a wide range of electrical coupling (see DISCUSSION).
NOISE AND COUPLING EFFECTS IN A NEURAL NETWORK
2603
previous simulation, noise from the other nine cells and oscillations prevented the cells from synchronizing to the input.
There was only one combination of noise and coupling tested
in which the 10 neurons synchronized to the input signal (Fig.
6, A and B). This occurred with high coupling (Cmax ⫽ 2.5) and
relatively low noise variance (1.36 pA2). Interestingly, in this
model, this noise level corresponds to the physiological baseline noise in a hippocampal slice (Sayer et al. 1989; Wahl et al.
1997). The neural array was very excitable at this combination
of coupling and noise: it oscillated at about 2 Hz without a
periodic signal present (Fig. 6A, top trace). This response
caused a low-amplitude, broad hump in the power spectrum
centered at ⬃2 Hz, whereas with addition of a 2.5-Hz signal, a
sharp spectral peak appeared at exactly 2.5 Hz. It is important
to note that signal frequency had no effect on signal detection
in any simulation up to this point. Even when the periodic
signal was raised above threshold, there was still no recruitment except at this specific combination (data not shown).
Oscillations corrupted the signal frequency at the next noise
level (Fig. 6A, third trace). Therefore in this network, a single
neuron cannot recruit the other cells unless they are very
excitable and prone to fire randomly near the input frequency.
The interesting ability of the periodic signal to recruit the
excitable network was further investigated by using a 6-Hz
input signal. This frequency was chosen after noting that the
network oscillated near 5.8 Hz with 17 pA2 noise and Cmax ⫽
0.9 without a periodic signal (Fig. 6, C and D). As predicted
above, the 6-Hz signal was able to synchronize the network
quite well. However, synchronization only occurred at the
specific combination of coupling and noise, demonstrated by
the unusual SNR plot. As with the prior example, the network
was recruitable only when it was very excitable and prone to
J Neurophysiol • VOL
fire at a mean frequency just below that of the periodic driving
signal. These results suggest that the network can be tuned to
detect, or be recruited by, certain periodic signals by adjustment of coupling and noise.
Effects of changing gmax of the noise input
The model was used to test the hypothesis that noise produced by minis of different quantal amplitude affects signal
detection differently. The three preceding simulations were all
repeated using noise mini amplitude of 1 nS. Because the mean
amplitude of noise events was higher, lower frequencies were
required to produce the same noise variance compared with
0.22-nS noise events (see DISCUSSION). There were differences
noted for all three instances when compared with the lower
amplitude simulations.
CHANGING NUMBER OF UNCOUPLED NEURONS. Increasing the
number of uncoupled detecting neurons increased the peak
SNR with N as in the previous case (Fig. 7A compared with
3C). However, compared with the 0.22-nS simulations, the
improvement in signal detection was not as pronounced. The
peak SNR was much smaller for 1 (13), 5 (79), and 10 (85)
cells. In addition, more noise variance was necessary to reach
peak detection, an effect that occurred for all simulations in
this section (note different x axis scale in Fig. 7 compared to
Figs. 3– 6). However, the SNR to the right of the peak followed
SR predictions more closely because the noise did not produce
as many oscillations.
Increasing coupling with
10 neurons and a global input improved overall signal detection (Fig. 7B compare with 4B), but the effect was less pro-
CHANGING COUPLING IN 10 NEURONS.
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FIG. 4. Effect of coupling in 10 cells with a global subthreshold
signal. A: raw data with a global signal for noise of 10.2 and 17
nA2 and 2 coupling levels. At higher noise (17 nA2), the oscillations overwhelmed the signal. Coupling synchronized the network
response, making the network elements fire closer together
(squares). Signals detected amid the noise in the uncoupled network (stars). Trace is for 5 s of data. B: SNR for various levels of
coupling. Data are fitted to the SR equation. Coupling improved
detection for low noise and corrupted it for high noise.
2604
W. C. STACEY AND D. M. DURAND
nounced than with 0.22-nS minis. Again, the peak SNR was
lower with 1-nS minis (162 for Cmax ⫽ 0.5), and more noise
variance was required to reach peak SNR. There was much less
shift of the curve on both sides of the peak, showing that the
sensitivity to both low and high levels of noise was not as
pronounced.
Ten coupled neurons and a
single input cell were extremely sensitive to quantal size of the
noise source. The network responded poorly to the single input
under all circumstances—the SNR never exceeded 10, was
maximal with 0 coupling, and had worse signal detection than
a single cell (Fig. 7C, compare with Figs. 5, B and C, and 6B).
The nine neighboring neurons were not recruited by the signal
cell but rather began to fire in response to the noise. With large
RECRUITMENT TO A SINGLE INPUT.
J Neurophysiol • VOL
amplitude noise events, therefore, this system was unable to
show any significant form of recruitment.
High noise variance evokes an intrinsic frequency in a
single cell
In many simulations described above, high levels of noise
produced SNR below the values estimated by the SR curve,
particularly with noise generated by 0.22-nS minis. The diminished SNR was due to spontaneous, repetitive firing of neurons
generated by the noise. A single neuron begins to fire repetitively when noise is raised above a certain threshold and the
cell switches from Poisson-distributed shot noise (Bekkers and
Stevens 1989; Brown et al. 1979; Cox and Lewis 1996) to
near-periodic firing in a phase transition (Fig. 8). In this model
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FIG. 5. SR and coupling effects for a single detecting cell. A: raw data. Top: poor detection with low noise and 0 coupling.
Middle: signal cell had greatly improved detection with coupling ⫽ 0.9. Bottom: input. Traces are for 5 s of data. Power spectra.
Top: 2.5-Hz spike is present, but SNR is small. Bottom: SNR is larger with 0.9 coupling. Arrow: input frequency of 2.5 Hz. This
and all power spectra in following figures are in logarithmic scale (square millivolts per hertz). B: SNR for low noise levels. For
coupling ⬍2.5, SNR increased with increasing coupling. Data for 0 and 0.9 coupling are fit to SR equation (0 coupling: dashed line;
0.9 coupling: solid line; fit included data from C). SNR of data shown in A (square). C: SNR for higher noise levels. Coupling
decreased signal detection as oscillations appeared. SNR for an uncoupled network was below that of a single cell due to noise from
the other 9 cells. SR equation fits are included for the response of a lone cell (dotted line) as well as 0 and 0.9 coupling. SNR of
data in D (square). D: raw data. Top: detection without coupling was similar to that in a single cell until the other 9 cells began
to fire. Middle: noise oscillations corrupt detection with 0.9 coupling. Bottom: input. Traces are 5 s. Power spectra. Top: SNR is
high without coupling. Bottom: signal frequency absent. Oscillations produce frequency hump approximately 4 Hz.
NOISE AND COUPLING EFFECTS IN A NEURAL NETWORK
2605
using the 0.22-nS noise source, the phase transition occurred
between 17 and 26 pA2 noise variance and began at a frequency of ⬃4 Hz. This mean intrinsic frequency increased and
became more discrete (a sharper spike in the frequency domain) with increasing noise variance. The response to a DC
current is included for comparison. These results are similar to
those found for autonomous SR in simpler neuron models
(Gang et al. 1993; Lee et al. 1998). The physiological noise
source in this model had a net depolarizing effect, but the
neural response herein is comparable to previous work with
zero mean noise (see DISCUSSION). No long-term accommodation effects occur because this model was designed to monitor
detection of individual signals and did not include the effects of
slower channels such as the afterhyperpolarization (AHP) potassium channel (Warman et al. 1994). This behavior of a
single neuron plays an important role when large amounts of
low-amplitude noise are present, especially when the neuron is
coupled to other cells.
J Neurophysiol • VOL
Noise and coupling contributions to bursting
The repetitive firing in a single cell was also present in the
network and created oscillatory behavior at high noise levels.
This effect was enhanced by coupling. The network was used
to test the hypothesis that noise and electrical coupling combine to produce synchronous bursts similar to those generated
by low calcium preparations (Perez Velasquez et al. 1994).
Ten cells were connected with various levels of coupling as
noise (0.22-nS source) was added to the system with no periodic signal input. The network response for varying coupling
and noise is shown in Fig. 9. With no coupling, all 10 cells fire
asynchronously. For very small amounts of coupling (Cmax ⫽
0.1), the response was asynchronous for low noise (17 pA2),
but increasing the noise (136 pA2) synchronized the network
(Fig. 9A). This level of coupling is very small, producing
⬍0.01% change in neighboring cells (measured as ⌬Vm(cell A)/
⌬Vm(cell B)). For higher levels of coupling, even baseline noise
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FIG. 6. Recruitment of the excitable network with high
coupling. A: raw data (2.5 coupling). Top: without signal, the
network fired near 2 Hz. Middle: with signal added, the
network became more periodic and fired at 2.5 Hz. Bottom:
at higher noise levels, the network fired faster than 2.5 Hz.
Traces are 5 s. Power spectra. Top: approximately 2 Hz
firing. Middle: sharp peak at 2.5 Hz. Bottom: sharp peak at
approximately 4.8 Hz. Signal frequency is buried in the
noise. B: same data as Fig. 5, stressing disparity in response
to 1.36 and 3.7 pA2 noise with 2.5 coupling. These data could
not be feasibly fit to the SR equation. SR fit for single cell
(dotted line). SNR of data in bottom 2 traces of A (squares).
C: raw data. Response to noise (17 nA2) without signal (top)
and with 6-Hz signal (bottom) with Cmax ⫽ 0.9. Trace is for
1.5 s of data. Power spectra. Top: mean frequency ⫽ 5.8 Hz.
Bottom: frequency peak at 6 Hz. Input frequency of 6 Hz
(arrow). D: SNR response to 6-Hz signal. The odd spike in
0.9 coupling occurs where the noise oscillations would be
near 5.8 Hz. In this condition the network was tuned to detect
a 6-Hz signal.
2606
W. C. STACEY AND D. M. DURAND
levels (1.7 pA2) produced synchrony (Fig. 9B). The amount of
synchrony (see Fig. 1D) for each combination of coupling and
noise is shown in Fig. 9C. The maximum synchrony occurred
for Cmax ⫽ 2.5 (the highest simulated) for 170 pA2. In all cases
the synchrony dropped as noise increased beyond 170 pA2.
Burst frequency was dependent on noise variance but not
coupling levels (Fig. 9D).
These results agree with previous simulations of CR (see
DISCUSSION). Because of the physiological design of the simu-
lation, these data also provide insight into the role that this
form of CR can play in a true neural system. Analysis of the
CR profile in light of physiology demonstrates two key conclusions about this system. First, the network will begin to
synchronize and fire even at resting noise levels (1.7 pA2) if
coupling is increased sufficiently. A single cell will not fire at
this noise level, requiring over 15 times higher noise variance
to fire periodically in response to noise (26 pA2, see Fig. 8).
The coupling makes the system much more susceptible to noise
FIG. 8. Response of a single cell to increasing noise Raw data. First: low noise produced random action potentials. Second:
increasing noise produced near-periodic firing. Third: as variance increased, firing became more periodic and faster. Fourth:
response to DC for comparison, 5-s traces. Power spectra. First: no appreciable frequency spikes for shot noise. Second: frequency
humps occur at ⬃4 Hz. Third: spikes become more discrete and more powerful for higher noise. Fourth: discrete spikes of DC
response.
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FIG. 7. Effect of changing amplitude of noise events. A:
effect of changing number of uncoupled detecting cells with
1-nS noise source (compare with Fig. 3). Data followed SR
equation very well. Increasing N increased peak SNR. More
noise variance was required to produce SR. B: effect of
coupling on 10 cells and global signal source (compare with
Fig. 4). SNR was improved for low noise and diminished at
high noise with increasing coupling, although not as much as
in Fig. 4. More noise variance was required for SR. C: SNR
plot: inability of network to recruit (see Figs. 5 and 6). SNR
⬍10 indicates inability to detect input well under any condition. Response of a single cell was better than all coupling
levels. Data point for raw data (square). Raw data: data
producing maximum SNR had multiple random action potentials from other 9 cells. Traces are 5 s.
NOISE AND COUPLING EFFECTS IN A NEURAL NETWORK
2607
oscillations, causing bursts even at baseline noise. Second, the
network can produce similar synchrony for many combinations
of coupling and noise. This phenomenon can be observed as
the isosynchronous topological clines in Fig. 9C, moving diagonally from high coupling/low noise to low coupling/high
noise. Therefore a system with low coupling can potentially
produce “high-coupled” synchrony merely by increasing the
noise in the system.
DISCUSSION
Choice of coupling mechanism
Coupling between CA1 cells can take the form of synaptic
connections or electrical coupling. Synaptic connections,
which have large amplitude and measurable delay, are very
sparse between CA1 neurons with a probability of connection
of 1/130 (Bernard and Wheal 1994) and require very large
networks to implement (Traub et al. 1999). In addition, they
also include inhibitory connections, which would introduce an
added degree of complexity and be difficult to treat as an
intrinsic property of the system without more precise in vitro
data. Electrical coupling can take the form of gap junctions or
ephaptic interactions, both of which are basic components of
J Neurophysiol • VOL
the CA1 layer (Andrews et al. 1982; Jefferys 1995; MacVicar
and Dudek 1981; Taylor and Dudek 1982; Vigmond et al.
1997). These two types of electrical coupling have quite similar effects. Both types allow bidirectional signals. The delay
for ephaptic connections and gap junctions is very small (Traub
et al. 1985). Both can also be involved in epileptiform activity
when increased (Dudek et al. 1986; Jefferys 1994; Perez
Velazquez and Carlen 2000). In the model presented here, both
types were combined into a general, delay-free current source.
The highest coupling levels simulated caused a depolarization of ⬍2 mV in response to an action potential in a coupled
cell. This depolarization is comparable with previous recordings of electrical coupling (Taylor and Dudek 1982; Vigmond
et al. 1997). High levels of coupling and noise evoked repetitive bursts of action potentials comparable with that in highly
coupled epilepsy models (Jensen and Yaari 1997; Perez
Velazquez et al. 1994; Schweitzer et al. 1992). Epileptiform
activity has been linked to increased electrical coupling (Dudek
et al. 1986; Jefferys 1994; Lee et al. 1995; Perez Velazquez and
Carlen 2000). The normal physiological range of coupling is
clearly below the level that causes constant bursting, but can be
increased by effects such as osmolarity changes (Schwartzkroin et al. 1998), and low Ca2⫹ (Bikson et al. 1999; Haas and
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FIG. 9. Noise oscillations and synchrony. A:
raw data (0.1 coupling). First: no response to low
noise. Second: cells fired asynchronously. Third:
cells synchronized even at this low coupling level
with high noise, 1-s traces. Power spectra. Second:
no discrete frequency. Third: frequency hump at
⬃7 Hz. B: raw data (2.5 coupling). First: bursting
at baseline noise level. Second: peak synchrony
level. Third: very high noise levels caused diminished synchrony. Power spectra. First: low frequency bursting formed a wide hump. Second:
peak synchrony occurred at ⬃8 Hz. Third: high
noise bursting at ⬃4.5 Hz. C: 3-dimensional plot
showing effects of coupling and noise on the synchrony of the network. Synchrony increased with
coupling and with noise, having maximum slope
on a diagonal when both are increased together.
Equal values of synchrony can be achieved for
many combinations of synchrony and noise. This
profile is typical of CR. D: frequency of oscillations changes with D, but there were only insignificant differences with Cmax of 0, 0.5, and 2.5.
2608
W. C. STACEY AND D. M. DURAND
Jefferys 1984), which also have been used as models of epileptiform activity. Therefore electrical coupling strongly affects signal detection in CA1 cells and suggests a pathophysiological role of the effects shown in this paper.
Noise characteristics affect stochastic resonance in neural
networks
Because of the time constant of neural membranes, the
membrane voltage cannot return to baseline if signals arrive in
quick succession. Noise produced by minis of lower amplitude
(lower gmax) requires higher mean firing frequency to produce
the same variance as higher amplitude minis, as shown in Fig.
2 and Eq. 5, which is based on the Poisson nature of the noise
(␴2 ⫽ variance, A ⫽ amplitude, f ⫽ mean frequency)
␴2 ⬀ A2 䡠 f
(5)
Noise oscillations and bursting activity in a CA1 network
When a single cell was presented with high levels of noise
it fired periodically as in autonomous SR (Gang et al. 1993;
Lee et al. 1998). When multiple cells were coupled, the network response was similar as the noise variance increased,
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Our data show that although similar variance can be produced
with minis of disparate amplitudes, the neural response can
vary significantly. There was a significant difference in the
response to noise produced by minis with gmax ⫽ 1 and 0.22
nS, which encompasses the range of published baseline noise
values (Bekkers and Stevens 1989; Destexhe et al. 1998;
McBain and Dingledine 1992; Stricker et al. 1996). The simulations showed that 0.22-nS minis produced better signal
detection (higher SNR, peak at lower variance compared with
1-nS minis) to the left of peak SNR and increased noise
oscillations beyond peak SNR. As a comparison, the response
to low-amplitude minis more closely approximates the response to a DC current: raising the voltage slightly changed the
threshold for subthreshold signals and improved detection;
raising it too much caused periodic firing. The response to
higher amplitude minis, conversely, is more similar to coincidence detection: temporal summation of discrete synaptic
events. SR is a useful tool that facilitates analysis of the system
under both conditions. This effect stresses the importance of
using models that address the frequency characteristics of
neurons as well as carefully addressing the relationship between amplitude and frequency of noise events in SR analysis.
Noise in CA1 can vary significantly from baseline in size
(Bekkers et al. 1990; Dobrunz and Stevens 1999; Larkman et
al. 1997) and frequency (Destexhe and Paré 1999; Kamondi et
al. 1998; Leung 1982). LTP can change both the amplitude and
frequency of events (Malgaroli and Tsien 1992; Stricker et al.
1996; Wyllie et al. 1994). The physiological range of mini
amplitude and frequency in these cells is therefore larger than
that simulated in this model. With an even larger range, it is
clear that the neural response will depend greatly on the noise
characteristics of the system. We conclude that it is important
to evaluate the specific characteristics of noise presented to the
cells rather than merely determine the internal noise variance.
This conclusion has implications in the study of SR in neurons
as well as in determining the characteristics leading to epileptiform bursting.
ranging from shot noise to near-periodic firing typical of CR
(Hu and Zhou 2000; Pham et al. 1998; Pradines et al. 1999;
Wang et al. 2000). This result was previously detailed in more
general mathematical models of neuronal networks (Kurrer and
Schulten 1995; Rappel and Karma 1996). It should be noted
that although our results agree in all respects tested with
previous work in CR, the noise source used herein was only
excitatory and thus differed from previous CR work using zero
mean noise. The effects of coupling on the frequency of
oscillators have been known for some time (Kepler et al. 1990).
Chaos has also been shown to decrease when comparable
systems become more coherent and periodic (Molgedey et al.
1992). By analyzing the role of noise and coupling with a
physiological model of hippocampal cells, these earlier mathematical analyses can now be applied to physiological situations.
The advantage of using this anatomic model is that it allows
modulation of physiological sources that can be compared
directly to experimental data. The input variance in Fig. 9C
ranges from 1.7 to 510 pA2. This range of synaptic current
produced a somatic voltage variance of 14,000 – 4,200,000
␮V2. The low end is a good estimate of noise in a quiescent cell
in the hippocampal slice (Wahl et al. 1997), but the higher
value is 300 times greater than baseline and must be analyzed
in view of the noise sources in CA1. We have previously
described how the in vivo inputs that are not present in the slice
can reasonably increase the background noise 300-fold (Stacey
and Durand 2001). However, it is clear that this high level of
noise would disrupt detection of all synaptic signals since all
cells would be firing repetitively, insensitive to any other
signal. Therefore these simulations deal with phenomena that
would be expected to occur within the region of physiological
noise levels.
The synchrony achieved with combinations of coupling and
noise is a demonstration of a modified form of coherence
resonance in a physiological model. It also produces synchronous, periodic bursting very similar to some models of epileptiform activity. With Cmax ⫽ 0 the output was uncorrelated, but
even small amounts of coupling produced some synchrony if
the noise level was high. Synchrony also increased with coupling. Maximum synchrony occurred when coupling was highest and noise was near 170 pA2 (Fig. 6C). This synergistic
relationship provides a means by which small networks can
reach synchrony for a wide range of coupling and noise. In a
network where either noise or coupling is low, increasing the
other parameter will increase synchrony. Even baseline noise
(12,000 ␮V2) or very small coupling (Cmax ⫽ 0.1) can be
brought to synchrony by this method. Conditions in which both
coupling and noise increase simultaneously cause the greatest
change. This finding agrees with experimental evidence that
noise is increased before an epileptiform burst (McBain and
Dingledine 1992) and also after stimulation (Manabe et al.
1992; Mennerick and Zorumski 1995) and coupling increases
in some forms of epileptic activity (Dudek et al. 1986; Jefferys
1994; Perez Velazquez and Carlen 2000). CR in this physiological model therefore provides an interesting insight into the
possible role of noise and coupling in the CNS: it is a simple
model of epileptic bursting. These results suggest that epileptiform activity can be evoked in a network by increasing
excitability through a combination of increased noise and coupling that separately would not be sufficient to cause bursting.
NOISE AND COUPLING EFFECTS IN A NEURAL NETWORK
Network contributions to signal detection
J Neurophysiol • VOL
tuned to detect and fire at a specific frequency. Physiologically,
this effect could used by higher brain centers to tune CA1 or
other coupled neurons to detect specific frequencies by merely
changing the variance of presynaptic firing.
CONCLUSION
In summary, by using a CA1 model previously tested experimentally, we were able to construct a network, determine
the effects of coupling and noise on signal detection, and relate
the results to physiological data. The network showed large
improvements in SNR as coupling and N increased for noise
near baseline, similar to other work in AESR. At high noise
levels, the SNR did not match values predicted by SR due to
periodic oscillations. These oscillations, as well as the specific
characteristics of the SNR, were strongly dependent on the
quantal amplitude of the noise, showing the importance of
addressing this parameter rather than merely the overall variance. The highly coupled network was very sensitive to specific driving signals and was able to synchronize to them. Thus
while SR does appear to be feasible under physiological conditions in this network, there are interesting deviations from SR
theory that may have deep implications in the CNS. These
results demonstrate the importance of relating neural AESR to
physiological parameters to understand its possible role in
vivo. CR in this model demonstrated the synergistic role of
coupling and noise in making the network synchronize, and is
a simple model of epileptiform activity as well as being the first
demonstration of coherence resonance in a vigorous physiological neural model. These studies demonstrate interesting
physiological implications of both coupling and noise on signal
detection in the brain, using the framework of CR and AESR
as investigative tools.
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