synaptic noise and physiological coupling generate high-frequency oscillations in a hippocampal...

doi: 10.1152/jn.00223.200288:2598-2611, 2002. ;J Neurophysiol

William C. Stacey and Dominique M. Durandin a Simulated Physiological Neural NetworkNoise and Coupling Affect Signal Detection and Bursting

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Noise and Coupling Affect Signal Detection and Bursting in aSimulated Physiological Neural Network

WILLIAM C. STACEY1,2 AND DOMINIQUE M. DURAND1

1Department of Biomedical Engineering, Case Western Reserve University; and 2Department of Neurology, UniversityHospitals of Cleveland, Cleveland, Ohio 44106

Received 27 March 2002; accepted in final form 9 July 2002

Stacey, William C. and Dominique M. Durand. Noise and couplingaffect signal detection and bursting in a simulated physiological neuralnetwork. J Neurophysiol 88: 2598–2611, 2002; 10.1152/jn.00223.2002.Signal detection in the CNS relies on a complex interaction betweenthe numerous synaptic inputs to the detecting cells. Two effects,stochastic resonance (SR) and coherence resonance (CR) have beenshown to affect signal detection in arrays of basic neuronal models.Here, an array of simulated hippocampal CA1 neurons was used totest the hypothesis that physiological noise and electrical coupling caninteract to modulate signal detection in the CA1 region of the hip-pocampus. The array was tested using varying levels of coupling andnoise with different input signals. Detection of a subthreshold signalin the network improved as the number of detecting cells increasedand as coupling was increased as predicted by previous studies in SR;however, the response depended greatly on the noise characteristicspresent and varied from SR predictions at times. Careful evaluation ofnoise characteristics may be necessary to form conclusions about therole of SR in complex systems such as physiological neurons. Thecoupled array fired synchronous, periodic bursts when presented withnoise alone. The synchrony of this firing changed as a function ofnoise and coupling as predicted by CR. The firing was very similar tocertain models of epileptiform activity, leading to a discussion of CRas a possible simple model of epilepsy. A single neuron was unable torecruit its neighbors to a periodic signal unless the signal was veryclose to the synchronous bursting frequency. These findings, whenviewed in comparison with physiological parameters in the hippocam-pus, suggest that both SR and CR can have significant effects onsignal processing in vivo.

I N T R O D U C T I O N

The effect of noise on detection of subthreshold signals inthreshold-detecting systems, or stochastic resonance (SR), hasbeen investigated in several physiological and nonphysiologi-cal systems. SR theory was originally developed to describe asingle bistable element (Benzi et al. 1981; Dykman et al. 1995;Moss et al. 1994) and was later expanded to describe a mono-stable system like a neuron (Stocks et al. 1993; Wiesenfeld etal. 1994). Both computer simulations and in vitro experimentshave been used to demonstrate SR behavior in a number ofneural systems (Braun et al. 1994; Bulsara et al. 1991; Collinset al. 1996; Douglass et al. 1993; Levin and Miller 1996; Pei etal. 1996; Russell et al. 1999). Recent studies in the CA1 layerof the hippocampus have indicated that SR can function to

influence signal detection in this system (Gluckman et al. 1996,1998; Stacey and Durand 2000) using endogenous noisesources (Stacey and Durand 2001). Even subtle changes aboveresting noise variance can improve the ability of neurons todetect subthreshold signals. The influence of noise on signaldetection in CA1 is therefore quite significant and relevant toinformation processing in the brain.

The SR relation between signal-to-noise ratio (SNR) andnoise for a neuron is given in Eq. 1. The SNR is 0 withoutadded noise, rises quickly to a maximum peak with the addi-tion of noise, and gradually decreases to 1 (D � noise intensity,� � signal strength, and �U � threshold barrier height) asnoise overcomes the output

SNR�� U

D�2

e��U/D (1)

Examples of this curve are presented in several of the laterfigures in this paper. SR has been found to have novel prop-erties on larger numbers of elements. SR in an array of ele-ments with a summed output generates an increased SNR to abroader range of noise (Chialvo et al. 1997; Collins et al.1995), although this simulated finding was not found experi-mentally in cochlear nerves (Morse and Roper 2000). Whenthe elements are coupled, as in array-enhanced SR (AESR), thesignal detection improvement can vary depending on how thecoupling is performed (Inchiosa and Bulsara 1995a,b; Kana-maru et al. 2001; Lindner et al. 1995; Locher et al. 1996). Inthis paper, we have chosen a coupling source to simulateelectrical connections between closely neighboring cells (seeMETHODS). AESR has important implications in the nervoussystem, because it can make a neural network sensitive to abroad range of input signals when noise is present (Stocks andMannella 2000).

Noise alone can produce oscillatory behavior in a singleelement under certain conditions, an effect closely related toSR (Gang et al. 1993; Lee et al. 1998). Another similar effectoccurs in coupled arrays of neural models that are not pre-sented with any periodic signal: they are able to respond withsynchronous, near-periodic outputs when noise is added (Huand Zhou 2000; Pham et al. 1998; Pradines et al. 1999; Wanget al. 2000). These effects have been called coherence reso-nance (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 paymentof page charges. The article must therefore be hereby marked ‘‘advertisement’’in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

J Neurophysiol88: 2598–2611, 2002; 10.1152/jn.00223.2002.

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used to describe only the case of a coupled network of neuronsand autonomous SR that of a single neuron. In CR, the abilityof noise to produce a synchronous, periodic response in thesystem is dependent on the noise characteristics and couplingbetween 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 synchronyfollows a bell-shaped curve as noise intensity increases(Neiman et al. 1997).

While SR in a single element has been verified experimen-tally using several different models, experimental evidencedemonstrating the presence or utility of AESR or CR in phys-iological neural systems is lacking. Although several differentmathematical neural models have been used to investigateAESR and CR, even simultaneously (Lindner and Schiman-sky-Geier 1999, 2000), there has been little correlation of theeffects with physiological parameters such as membrane dy-namics, channels, and voltage. Rationalization and evaluationof the simulated data in light of its physiological ramificationshas been difficult without these analyses or experimental ver-ification. It is a major goal of this work to provide suchcorrelations to physiological data by using a neural model thathas been verified experimentally and that utilizes cellular pa-rameters.

The beneficial effect of noise on signal detection has intrigu-ing implications in the brain. Neurons such as CA1 pyramidalcells in the hippocampus are responsible for integration ofsignals from many different sources (Andersen 1990). Signaldetection in these cells is complicated by the presence of noise(Bekkers et al. 1990; Destexhe and Pare 1999; Kamondi et al.1998; Sayer et al. 1989;), the large number of synapses, anddendritic attenuation (Spruston et al. 1993), a situation thatmakes them ideal candidates for physiological SR (Stacey andDurand 2000, 2001). It is quite possible that AESR and CRalso have physiological effects in the CA1 layer due to thenetwork organization. The presence of these phenomena in thebrain would have broad implications for normal signal detec-tion as well as for abnormal detection such as epileptogenesis.In this paper, we seek to analyze the effects of noise andcoupling on signal detection in an array of simulated CA1cells. This simulation is based on a detailed implementation ofspecific cellular parameters in CA1 cells (Warman et al. 1994)that have been verified experimentally (Stacey and Durand2000, 2001). Noise and coupling have been added according tophysiological conditions and have been carefully verified withexperimental data. The model is used to test the hypothesis thatnoise and coupling can work synergistically to improve detec-tion of a signal in the CA1 layer of the rat hippocampusthrough the effects of AESR and CR.

M E T H O D S

Network simulation

An array of 10 simulated CA1 neurons previously described (Sta-cey and Durand 2000, 2001) was implemented using NEURONsoftware (Hines 1993). The neurons (Fig. 1A) had membrane param-eters derived from experimental data (Table 1) and were identical toeach 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 atthe synapses were recorded separately with a sampling rate of 2,000 Hz.

Synaptic events generating the periodic input signal and randomevents were simulated using AMPA synapse functions (Destexhe etal. 1998). The amplitude of synaptic events was modulated by chang-ing the maximum conductance (gmax) of the AMPA current. Theperiodic signal input was positioned on the distal branch of the apicaldendrite of each cell and the amplitude adjusted to a subthresholdlevel (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 wereable to return to baseline between pulses to avoid any frequency-dependent effects of the signal. The signal was applied either to allneurons simultaneously (“global input”) or to one neuron alone (“sin-gle input”; Fig. 1B). Noise was added to simulate any synaptic activitythat is uncorrelated to the periodic signal. The noise was generated by55 presynaptic axons, 10 of which contacted between three and ninedifferent CA1 cells as en passant axons (e.g., Fig. 1C), while theremainder made independent connections. The en passant connectionswere generated randomly and did not cause any significant correlationamong the CA1 cells. This connection scheme simulates the connec-tivity that exists in a rat hippocampal slice (Andersen 1990), whereneighboring CA1 cells often receive the same presynaptic signal(Bernard and Wheal 1994).

Noise

Simulations were performed for two values of maximum conduc-tance (gmax) at the noise synapses: 0.22 and 1 nS. These two valuesencompass the experimental range of miniature excitatory postsynap-tic 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) wasmodulated by changing the mean firing frequency of the noise syn-apses. The firing of each presynaptic axon, which evoked a synapticevent, was calculated independently by comparing a threshold with auniform random number generated at each time step (Stacey andDurand 2000). Changing this threshold thus changed the probabilityof firing at each time step. The characteristics of the synaptic noisecurrent are shown in Fig. 2, which spans the entire range of noisesimulated. Noise bandwidth and spectral power increased as expectedwith increased firing probability. Increasing D increased the meandepolarizing current in a nearly linear fashion. The noise source hadsome low-pass filtering effects due to the frequency characteristics ofthe AMPA synapse, but they were present for much higher levels ofD than those used in these simulations. Also of note is the differentresponse to noise with gmax � 0.22 nS versus 1 nS (see DISCUSSION).Noise was locally uncorrelated (Lindner et al. 1995) except the minoreffect from synapses connected by en passant axons. There was notime delay between the multiple synapses contacted by the en passantaxons. Synaptic noise variance was computed from recordings of thesynaptic current.

Coupling

Coupling between each pair of cells was modeled as an additionalvoltage controlled current source (I coupling

i ) in the soma of each cell(Eq. 2). Current effects from all neighboring neurons were summed toproduce a single current input at the soma

Icouplingi � �

j�0

N�1

a � cij�Vj � Erest� �j � i� (2)

fC�cij� �1

Cmax � 0all cij � cji (3)

The current amplitude was determined by the driving potential of thecoupled neuron (Vj � Erest) and a coupling weight (cij). The couplingweight, cij, was bidirectional and randomly assigned from the uniform

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interval (0 3 Cmax), as shown by the probability density function inEq. 3. In other words, each cell “i” received a coupling current fromeach of the nine cells “j”, and the weight of coupling current wasrandomized on a scale determined by the value of Cmax. The couplingstrength was multiplied by the arbitrary constant “a” to allow simpleimplementation in NEURON. The coupled current caused the poten-tial to move in the same direction as the driving potential. This methodproduced an instantaneous change in somatic current that was depen-dent on the voltage change of another cell, which is similar to

physiological forms of electrical coupling (see DISCUSSION). The dis-tribution did not take into account any spatial effects suggested by thelinear schematization in Fig. 1, B and C. The coupling mechanism didnot generate any shunting between the cells that would diminish thesignal strength (Lindner et al. 1995).

Signal analysis

The membrane voltages from all neurons were summed to producea network voltage response: Vsum � �

nVi (Collins et al. 1995). These

data were mean-detrended (Vsum(rest) � 0 mV) to remove the restingvoltage contribution from each cell. Any subthreshold voltages (Vsum

� 70 mV) were then masked to a value of 0 while preserving theamplitude of suprathreshold responses. In this manner, the only non-zero data contained in the output was from action potentials andpreserved their net amplitude above resting voltage to determine whenmultiple cells were firing. The resulting output time series was used toobtain the power spectrum (Wiesenfeld and Moss 1995) using thePSD function in Matlab (50,000-point Hanning window; 45,000-pointoverlap). The power at the input signal frequency (2.5 Hz) wasdivided by the baseband power (noise) in the region near 2.5 Hz toobtain the SNR. All data in this paper are averages of three indepen-dent simulations. The SNR obtained from each value of noise variancewas plotted and used as data for fitting to the SR equation (Stacey andDurand 2000). In multiple-trace plots the SD is omitted from sometraces for clarity, but in all cases the omitted deviations were compa-rable 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 tocompare the SR responses.

For calculation of coherence, the output of a network presented

FIG. 1. CA1 model and network organization. A: sche-matic of an individual CA1 cell, showing positions of noisesources (stars) and subthreshold signal (S). Parameters of theneural segments are in Table 1. B: linear representation of thenetwork, showing how the global signal source axon con-tacted all 10 cells, and the single source only contacted thecell at the far left, the “signal cell.” The network was imple-mented as a randomly arranged cluster: the spatial organiza-tion implied by the schematization was not included in themodel. C: implementation of the network. Diagram shows 2of the en passant noise axons that contacted multiple cells.Membrane voltage of each cell was summed to produce theoutput of the network. D: calculation of synchrony. Powerspectrum of the system output produced frequency spikeswhen coherence resonance (CR) was present. Synchrony (�)was evaluated for each simulation independently (Eq. 4).

TABLE 1. CA1 model specifications

DendriteRm, �cm2 14,000Cm, �F/cm2 2

SomaRm, �cm2 28,000Cm, �F/cm2 1

Ri, �cm 150

Diam(�m) L (�m) Segments �

DimensionsSoma 15 20 1 0.011Basal 4 250 5 0.2591° Apical 3.5 400 10 0.443Auxiliary 2° apical 3.13 400 5 0.468Main 2° apical 1 400 5 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 variancewas modulated by changing the mean frequency of firing.

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with noise but no periodic signal was evaluated. The power spectrumof each simulation was computed as described above and then ana-lyzed individually. Noise-induced synchrony appeared as a spike inthe power spectrum. Synchrony from this resonant peak was measuredusing the coherence factor � in Eq. 4, where h is the height of thepeak, p is the frequency of the peak, and � is the width at half-peakheight (Gang et al. 1993)

� � h � p/� (4)

R E S U L T S

The array of neurons was configured in four basic forms.First, the number of neurons in the network was varied withoutany coupling as the subthreshold signal and noise were pre-sented to all neurons simultaneously. Second, with 10 neurons,coupling was varied as all neurons received the signal andnoise again. These two configurations were designed to test theeffect of varying coupling and the number of neurons ondetection of a global signal. Third, the second simulation wasrepeated with signal removed from nine of the neurons. Thisconfiguration tested the ability of noise and coupling to affectrecruitment of a network to a signal present at only one cell.Results were first obtained for these three configurations usingnoise with gmax � 0.22 nS and then repeated for 1 nS. Fourth,the signal was removed entirely and the 10-neuron networkwas presented with varying degrees of coupling and noise(0.22-nS source). This configuration was used to investigateoscillatory behavior that was present in many of the previoussimulations.

Effect of multiple uncoupled neurons on subthreshold signaldetection

The network was first used to test the effect of the numberof detecting neurons on detection of subthreshold signals.Simulations with 1, 3, 5, and 10 neurons were configured withglobal inputs without coupling. Noise with mini-amplitude of0.22 nS was applied. Several levels of noise variance wereadded to the system along with the subthreshold signal. Theresponse shows that noise can improve the network’s detectionof the subthreshold signal (Fig. 3A). As the number of neurons

increased, the amplitude of the response increased and detec-tion of the input signal was improved (Fig. 3B). Increasingnoise variance produced the typical SR curve and peak SNRincreased as more cells were added (Fig. 3C). Addition of noiseto the system produced peak SNR values of 174, 245, and 467for 1, 3, and 10 cells, respectively. SNR for 5 cells werebetween those for 3 and 10 cells at all noise levels (data notshown). For variance �15 pA2, the SNR was below the levelpredicted by SR. This was due to repetitive firing (oscillations)of the network present at high noise variance, which increasedthe background noise and will be discussed later. The center ofthe SR peak did not change with the number of neurons, but theamplitude varied with N. The change corresponded to anincreasing 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 improvedetection of a global, subthreshold signal for low noise levels,but detection can be poorer than predicted by SR theory at highnoise variance for noise with quantal amplitude on the low endof physiological range.

Effect of coupling on signal detection

The network was then used to test the effects of electricalcoupling between cells when all neurons received the sub-threshold input. Coupling was added and modulated betweencells in the 10-neuron array with the same noise source asbefore. The network was tested with the global input config-uration with four levels of Cmax: 0, 0.1, 0.3, and 0.5. Theselevels were chosen to span a spectrum of physiological cou-pling. The maximum coupling (cij � 0.5) produced a 0.5-mVdepolarization 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 detec-tion of a subthreshold signal in the presence of noise is shownin Fig. 4A. For noise (10.2 pA2) that produced peak SNR, in theuncoupled network the signal was detected but the latencybetween the first and last cell to fire was often �20 ms (firsttrace). In the coupled network, there was a more coherentresponse: all 10 neurons tended to fire within a 10-ms span

FIG. 2. Noise characteristics. A: mean was calcu-lated for the current at 1 AMPA noise synapse for therange of noise variances used in the simulations. Thenoise source behaved differently to the 2 noise sig-nals due to characteristics of the AMPA synapse.Higher levels of variance were required for the 1-nSsimulations (see RESULTS). B: power spectrum: datafor 0.22 nS noise source with 2 mean firing frequen-cies. Bandwidth and spectral power increased for the400-Hz case. Raw data: current from the synapse foreach frequency. Trace is 0.5 s.





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(second trace). With high levels of noise (17 pA2), extraneousaction potentials were generated, an effect worsened by in-creased coupling (third and fourth traces). Thus small levels ofcoupling and noise improved the signal detection, but as bothincreased the network fired clustered action potentials thatoverwhelmed the input signal. Results from 0.1 and 0.3 cou-pling levels (not shown) had a similar relationship, with valueslying between the data shown for Cmax � 0 and 0.5.

The SNR plots show the effect of coupling on signal detec-tion (Fig. 4B). Coupling significantly increased the networksensitivity to noise, as shown by an increased SNR at lowernoise levels and a sharper decline in SNR after peak detection.The peak SNR for maximum coupling (Cmax � 0.5) wassignificantly higher than that with zero coupling (1,208 vs.467) and occurred at lower noise (5.1 vs. 13.6 pA2). Thecoupling therefore caused a shift in the SR curve up and to theleft (decreased �U, increased �). However, with higher noisevariance (17 pA2), the SNR was greatly decreased for higher

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 signaldetection and SR in the network. As the coupling rises, thenetwork becomes increasingly more sensitive to noise—bothto improve detection with low noise variance and hamperdetection at higher variance.

Effect of noise and coupling on signal detection by a singleneuron in a network

The simulation was then performed with the single inputconfiguration (Fig. 1B) to test the hypothesis that a single cellcan detect a subthreshold signal and recruit its neighboringcells through the combined effects of SR and coupling. Valuesof Cmax � 0, 0.1, 0.3, 0.5, 0.7, 0.9, 2.5, and 3 were chosen toencompass 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 ac-tion potential was 1.8 mV (cij � 2.5). Note that, due torandomization, the values of cij for Cmax � 3 were usually2.5. This situation is representative of a single neuron re-ceiving a signal surrounded by several quiescent neurons witha wide range of electrical coupling (see DISCUSSION).

SIGNAL DETECTION OF NETWORK WITH LOW NOISE VARIANCE.

The network response to the single input at low noise levels isshown in Fig. 5A. For low-to-moderate coupling (Cmax � 0.9),the network did not synchronize to the signal. The signal cellwas 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 thatincreased 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 noiselevels is included. These results show that even small couplinglevels are able to improve a cell’s ability to detect a subthresh-old signal in a noisy system through SR.

SIGNAL DETECTION OF NETWORK WITH HIGH NOISE VARIANCE.

Whereas the network’s improved sensitivity to noise was ben-eficial for low noise, it greatly diminished SNR at higherlevels. Figure 5, C and D, shows that increased couplingproduced oscillations that corrupt signal detection and departfrom the SR equation. For all levels of coupling tested, theSNR was once more below the predicted SR value at highnoise. An additional effect occurred independent of noise os-cillations: the added noise from the nine other cells also de-creased SNR. For this reason, detection with zero coupling wasworse than for a single cell as noise increased. The driving cellis therefore able to use coupling to enhance its own signaldetection at low noise, but is normally unable to overcome theadditional 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 ofthe single cell to recruit its neighbors to the subthresholdsignal. Recruitment was defined as more than one cell firing inresponse to the input signal and producing an increased SNR.Under most circumstances, there was no significant recruit-ment in this network, as seen by the action potentials firingindependently and the low SNR values in Fig. 5. As in the

FIG. 3. Effect of number of cells in the network. A: top: raw data showingthe trace of a single cell with a subthreshold signal. Bottom: input signal. Whennoise was applied, the signal was detected. B: raw data. As in A, there was noresponse before the addition of noise (data not shown). As the number of cellsincreased, there was a concurrent increase in both the number of actionpotentials in phase with the signal and random action potentials. However, theincreased power in phase with the signal was stronger, leading to an improvedsignal-to-noise ratio (SNR). Note that the output was the sum of the array, somultiple concurrent action potentials appear as spikes �100 mV. Trace is for5 s of data. C: effect of number of cells on signal detection. Data are fit to thestochastic resonance (SR) equation. Increasing the number of detecting ele-ments changed the peak without changing the resonant center. All data strayedbelow SR theory at high noise. Bars indicate 1 SD for the 10-cell data.Comparable bars on the other data were removed for clarity.





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previous simulation, noise from the other nine cells and oscil-lations prevented the cells from synchronizing to the input.There was only one combination of noise and coupling testedin which the 10 neurons synchronized to the input signal (Fig.6, A and B). This occurred with high coupling (Cmax � 2.5) andrelatively low noise variance (1.36 pA2). Interestingly, in thismodel, this noise level corresponds to the physiological base-line noise in a hippocampal slice (Sayer et al. 1989; Wahl et al.1997). The neural array was very excitable at this combinationof coupling and noise: it oscillated at about 2 Hz without aperiodic signal present (Fig. 6A, top trace). This responsecaused a low-amplitude, broad hump in the power spectrumcentered at 2 Hz, whereas with addition of a 2.5-Hz signal, asharp spectral peak appeared at exactly 2.5 Hz. It is importantto note that signal frequency had no effect on signal detectionin any simulation up to this point. Even when the periodicsignal was raised above threshold, there was still no recruit-ment except at this specific combination (data not shown).Oscillations corrupted the signal frequency at the next noiselevel (Fig. 6A, third trace). Therefore in this network, a singleneuron cannot recruit the other cells unless they are veryexcitable and prone to fire randomly near the input frequency.

The interesting ability of the periodic signal to recruit theexcitable network was further investigated by using a 6-Hzinput signal. This frequency was chosen after noting that thenetwork oscillated near 5.8 Hz with 17 pA2 noise and Cmax �0.9 without a periodic signal (Fig. 6, C and D). As predictedabove, the 6-Hz signal was able to synchronize the networkquite well. However, synchronization only occurred at thespecific combination of coupling and noise, demonstrated bythe unusual SNR plot. As with the prior example, the networkwas recruitable only when it was very excitable and prone to

fire at a mean frequency just below that of the periodic drivingsignal. These results suggest that the network can be tuned todetect, or be recruited by, certain periodic signals by adjust-ment of coupling and noise.

Effects of changing gmax of the noise input

The model was used to test the hypothesis that noise pro-duced by minis of different quantal amplitude affects signaldetection differently. The three preceding simulations were allrepeated using noise mini amplitude of 1 nS. Because the meanamplitude of noise events was higher, lower frequencies wererequired to produce the same noise variance compared with0.22-nS noise events (see DISCUSSION). There were differencesnoted for all three instances when compared with the loweramplitude simulations.

CHANGING NUMBER OF UNCOUPLED NEURONS. Increasing thenumber of uncoupled detecting neurons increased the peakSNR with N as in the previous case (Fig. 7A compared with3C). However, compared with the 0.22-nS simulations, theimprovement in signal detection was not as pronounced. Thepeak SNR was much smaller for 1 (13), 5 (79), and 10 (85)cells. In addition, more noise variance was necessary to reachpeak detection, an effect that occurred for all simulations inthis section (note different x axis scale in Fig. 7 compared toFigs. 3–6). However, the SNR to the right of the peak followedSR predictions more closely because the noise did not produceas many oscillations.

CHANGING COUPLING IN 10 NEURONS. Increasing coupling with10 neurons and a global input improved overall signal detec-tion (Fig. 7B compare with 4B), but the effect was less pro-

FIG. 4. Effect of coupling in 10 cells with a global subthresholdsignal. A: raw data with a global signal for noise of 10.2 and 17nA2 and 2 coupling levels. At higher noise (17 nA2), the oscilla-tions overwhelmed the signal. Coupling synchronized the networkresponse, making the network elements fire closer together(squares). Signals detected amid the noise in the uncoupled net-work (stars). Trace is for 5 s of data. B: SNR for various levels ofcoupling. Data are fitted to the SR equation. Coupling improveddetection for low noise and corrupted it for high noise.





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nounced than with 0.22-nS minis. Again, the peak SNR waslower with 1-nS minis (162 for Cmax � 0.5), and more noisevariance was required to reach peak SNR. There was much lessshift of the curve on both sides of the peak, showing that thesensitivity to both low and high levels of noise was not aspronounced.

RECRUITMENT TO A SINGLE INPUT. Ten coupled neurons and asingle input cell were extremely sensitive to quantal size of thenoise source. The network responded poorly to the single inputunder all circumstances—the SNR never exceeded 10, wasmaximal with 0 coupling, and had worse signal detection thana single cell (Fig. 7C, compare with Figs. 5, B and C, and 6B).The nine neighboring neurons were not recruited by the signalcell but rather began to fire in response to the noise. With large

amplitude noise events, therefore, this system was unable toshow any significant form of recruitment.

High noise variance evokes an intrinsic frequency in asingle cell

In many simulations described above, high levels of noiseproduced SNR below the values estimated by the SR curve,particularly with noise generated by 0.22-nS minis. The dimin-ished SNR was due to spontaneous, repetitive firing of neuronsgenerated by the noise. A single neuron begins to fire repeti-tively when noise is raised above a certain threshold and thecell switches from Poisson-distributed shot noise (Bekkers andStevens 1989; Brown et al. 1979; Cox and Lewis 1996) tonear-periodic firing in a phase transition (Fig. 8). In this model

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. Thisand all power spectra in following figures are in logarithmic scale (square millivolts per hertz). B: SNR for low noise levels. Forcoupling 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. Couplingdecreased signal detection as oscillations appeared. SNR for an uncoupled network was below that of a single cell due to noise fromthe 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 ofdata in D (square). D: raw data. Top: detection without coupling was similar to that in a single cell until the other 9 cells beganto fire. Middle: noise oscillations corrupt detection with 0.9 coupling. Bottom: input. Traces are 5 s. Power spectra. Top: SNR ishigh without coupling. Bottom: signal frequency absent. Oscillations produce frequency hump approximately 4 Hz.





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using the 0.22-nS noise source, the phase transition occurredbetween 17 and 26 pA2 noise variance and began at a fre-quency of 4 Hz. This mean intrinsic frequency increased andbecame more discrete (a sharper spike in the frequency do-main) with increasing noise variance. The response to a DCcurrent is included for comparison. These results are similar tothose found for autonomous SR in simpler neuron models(Gang et al. 1993; Lee et al. 1998). The physiological noisesource in this model had a net depolarizing effect, but theneural response herein is comparable to previous work withzero mean noise (see DISCUSSION). No long-term accommoda-tion effects occur because this model was designed to monitordetection of individual signals and did not include the effects ofslower channels such as the afterhyperpolarization (AHP) po-tassium channel (Warman et al. 1994). This behavior of asingle neuron plays an important role when large amounts oflow-amplitude noise are present, especially when the neuron iscoupled to other cells.

Noise and coupling contributions to bursting

The repetitive firing in a single cell was also present in thenetwork and created oscillatory behavior at high noise levels.This effect was enhanced by coupling. The network was usedto test the hypothesis that noise and electrical coupling com-bine to produce synchronous bursts similar to those generatedby low calcium preparations (Perez Velasquez et al. 1994).

Ten cells were connected with various levels of coupling asnoise (0.22-nS source) was added to the system with no peri-odic signal input. The network response for varying couplingand noise is shown in Fig. 9. With no coupling, all 10 cells fireasynchronously. 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, producing0.01% change in neighboring cells (measured as �Vm(cell A)/�Vm(cell B)). For higher levels of coupling, even baseline noise

FIG. 6. Recruitment of the excitable network with highcoupling. A: raw data (2.5 coupling). Top: without signal, thenetwork fired near 2 Hz. Middle: with signal added, thenetwork 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 Hzfiring. Middle: sharp peak at 2.5 Hz. Bottom: sharp peak atapproximately 4.8 Hz. Signal frequency is buried in thenoise. B: same data as Fig. 5, stressing disparity in responseto 1.36 and 3.7 pA2 noise with 2.5 coupling. These data couldnot 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 for1.5 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 in0.9 coupling occurs where the noise oscillations would benear 5.8 Hz. In this condition the network was tuned to detecta 6-Hz signal.





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levels (1.7 pA2) produced synchrony (Fig. 9B). The amount ofsynchrony (see Fig. 1D) for each combination of coupling andnoise is shown in Fig. 9C. The maximum synchrony occurredfor Cmax � 2.5 (the highest simulated) for 170 pA2. In all casesthe synchrony dropped as noise increased beyond 170 pA2.Burst frequency was dependent on noise variance but notcoupling levels (Fig. 9D).

These results agree with previous simulations of CR (seeDISCUSSION). Because of the physiological design of the simu-

lation, these data also provide insight into the role that thisform of CR can play in a true neural system. Analysis of theCR profile in light of physiology demonstrates two key con-clusions about this system. First, the network will begin tosynchronize and fire even at resting noise levels (1.7 pA2) ifcoupling is increased sufficiently. A single cell will not fire atthis noise level, requiring over 15 times higher noise varianceto fire periodically in response to noise (26 pA2, see Fig. 8).The coupling makes the system much more susceptible to noise

FIG. 7. Effect of changing amplitude of noise events. A:effect of changing number of uncoupled detecting cells with1-nS noise source (compare with Fig. 3). Data followed SRequation very well. Increasing N increased peak SNR. Morenoise variance was required to produce SR. B: effect ofcoupling on 10 cells and global signal source (compare withFig. 4). SNR was improved for low noise and diminished athigh noise with increasing coupling, although not as much asin Fig. 4. More noise variance was required for SR. C: SNRplot: inability of network to recruit (see Figs. 5 and 6). SNR10 indicates inability to detect input well under any con-dition. Response of a single cell was better than all couplinglevels. Data point for raw data (square). Raw data: dataproducing maximum SNR had multiple random action po-tentials from other 9 cells. Traces are 5 s.

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: frequencyhumps occur at 4 Hz. Third: spikes become more discrete and more powerful for higher noise. Fourth: discrete spikes of DCresponse.





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oscillations, causing bursts even at baseline noise. Second, thenetwork can produce similar synchrony for many combinationsof coupling and noise. This phenomenon can be observed asthe isosynchronous topological clines in Fig. 9C, moving di-agonally from high coupling/low noise to low coupling/highnoise. Therefore a system with low coupling can potentiallyproduce “high-coupled” synchrony merely by increasing thenoise in the system.

D I S C U S S I O N

Choice of coupling mechanism

Coupling between CA1 cells can take the form of synapticconnections or electrical coupling. Synaptic connections,which have large amplitude and measurable delay, are verysparse between CA1 neurons with a probability of connectionof 1/130 (Bernard and Wheal 1994) and require very largenetworks to implement (Traub et al. 1999). In addition, theyalso include inhibitory connections, which would introduce anadded degree of complexity and be difficult to treat as anintrinsic property of the system without more precise in vitrodata. Electrical coupling can take the form of gap junctions orephaptic interactions, both of which are basic components of

the CA1 layer (Andrews et al. 1982; Jefferys 1995; MacVicarand Dudek 1981; Taylor and Dudek 1982; Vigmond et al.1997). These two types of electrical coupling have quite sim-ilar effects. Both types allow bidirectional signals. The delayfor ephaptic connections and gap junctions is very small (Traubet al. 1985). Both can also be involved in epileptiform activitywhen increased (Dudek et al. 1986; Jefferys 1994; PerezVelazquez and Carlen 2000). In the model presented here, bothtypes were combined into a general, delay-free current source.

The highest coupling levels simulated caused a depolariza-tion of 2 mV in response to an action potential in a coupledcell. This depolarization is comparable with previous record-ings of electrical coupling (Taylor and Dudek 1982; Vigmondet al. 1997). High levels of coupling and noise evoked repeti-tive bursts of action potentials comparable with that in highlycoupled epilepsy models (Jensen and Yaari 1997; PerezVelazquez et al. 1994; Schweitzer et al. 1992). Epileptiformactivity has been linked to increased electrical coupling (Dudeket al. 1986; Jefferys 1994; Lee et al. 1995; Perez Velazquez andCarlen 2000). The normal physiological range of coupling isclearly below the level that causes constant bursting, but can beincreased by effects such as osmolarity changes (Schwartz-kroin et al. 1998), and low Ca2� (Bikson et al. 1999; Haas and

FIG. 9. Noise oscillations and synchrony. A:raw data (0.1 coupling). First: no response to lownoise. Second: cells fired asynchronously. Third:cells synchronized even at this low coupling levelwith high noise, 1-s traces. Power spectra. Second:no discrete frequency. Third: frequency hump at7 Hz. B: raw data (2.5 coupling). First: burstingat baseline noise level. Second: peak synchronylevel. Third: very high noise levels caused dimin-ished synchrony. Power spectra. First: low fre-quency bursting formed a wide hump. Second:peak synchrony occurred at 8 Hz. Third: highnoise bursting at 4.5 Hz. C: 3-dimensional plotshowing effects of coupling and noise on the syn-chrony of the network. Synchrony increased withcoupling and with noise, having maximum slopeon a diagonal when both are increased together.Equal values of synchrony can be achieved formany combinations of synchrony and noise. Thisprofile is typical of CR. D: frequency of oscilla-tions changes with D, but there were only insig-nificant differences with Cmax of 0, 0.5, and 2.5.





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Jefferys 1984), which also have been used as models of epi-leptiform activity. Therefore electrical coupling strongly af-fects signal detection in CA1 cells and suggests a pathophys-iological role of the effects shown in this paper.

Noise characteristics affect stochastic resonance in neuralnetworks

Because of the time constant of neural membranes, themembrane voltage cannot return to baseline if signals arrive inquick succession. Noise produced by minis of lower amplitude(lower gmax) requires higher mean firing frequency to producethe 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 � f (5)

Our data show that although similar variance can be producedwith minis of disparate amplitudes, the neural response canvary significantly. There was a significant difference in theresponse to noise produced by minis with gmax � 1 and 0.22nS, which encompasses the range of published baseline noisevalues (Bekkers and Stevens 1989; Destexhe et al. 1998;McBain and Dingledine 1992; Stricker et al. 1996). The sim-ulations showed that 0.22-nS minis produced better signaldetection (higher SNR, peak at lower variance compared with1-nS minis) to the left of peak SNR and increased noiseoscillations beyond peak SNR. As a comparison, the responseto low-amplitude minis more closely approximates the re-sponse to a DC current: raising the voltage slightly changed thethreshold for subthreshold signals and improved detection;raising it too much caused periodic firing. The response tohigher amplitude minis, conversely, is more similar to coinci-dence detection: temporal summation of discrete synapticevents. SR is a useful tool that facilitates analysis of the systemunder both conditions. This effect stresses the importance ofusing models that address the frequency characteristics ofneurons as well as carefully addressing the relationship be-tween 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 etal. 1997) and frequency (Destexhe and Pare 1999; Kamondi etal. 1998; Leung 1982). LTP can change both the amplitude andfrequency of events (Malgaroli and Tsien 1992; Stricker et al.1996; Wyllie et al. 1994). The physiological range of miniamplitude and frequency in these cells is therefore larger thanthat simulated in this model. With an even larger range, it isclear that the neural response will depend greatly on the noisecharacteristics of the system. We conclude that it is importantto evaluate the specific characteristics of noise presented to thecells rather than merely determine the internal noise variance.This conclusion has implications in the study of SR in neuronsas well as in determining the characteristics leading to epilep-tiform bursting.

Noise oscillations and bursting activity in a CA1 network

When a single cell was presented with high levels of noiseit fired periodically as in autonomous SR (Gang et al. 1993;Lee et al. 1998). When multiple cells were coupled, the net-work response was similar as the noise variance increased,

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 moregeneral mathematical models of neuronal networks (Kurrer andSchulten 1995; Rappel and Karma 1996). It should be notedthat although our results agree in all respects tested withprevious work in CR, the noise source used herein was onlyexcitatory and thus differed from previous CR work using zeromean noise. The effects of coupling on the frequency ofoscillators have been known for some time (Kepler et al. 1990).Chaos has also been shown to decrease when comparablesystems become more coherent and periodic (Molgedey et al.1992). By analyzing the role of noise and coupling with aphysiological model of hippocampal cells, these earlier math-ematical analyses can now be applied to physiological situa-tions.

The advantage of using this anatomic model is that it allowsmodulation of physiological sources that can be compareddirectly to experimental data. The input variance in Fig. 9Cranges from 1.7 to 510 pA2. This range of synaptic currentproduced a somatic voltage variance of 14,000–4,200,000�V2. The low end is a good estimate of noise in a quiescent cellin the hippocampal slice (Wahl et al. 1997), but the highervalue is 300 times greater than baseline and must be analyzedin view of the noise sources in CA1. We have previouslydescribed how the in vivo inputs that are not present in the slicecan reasonably increase the background noise 300-fold (Staceyand Durand 2001). However, it is clear that this high level ofnoise would disrupt detection of all synaptic signals since allcells would be firing repetitively, insensitive to any othersignal. Therefore these simulations deal with phenomena thatwould be expected to occur within the region of physiologicalnoise levels.

The synchrony achieved with combinations of coupling andnoise is a demonstration of a modified form of coherenceresonance in a physiological model. It also produces synchro-nous, periodic bursting very similar to some models of epilep-tiform activity. With Cmax � 0 the output was uncorrelated, buteven small amounts of coupling produced some synchrony ifthe noise level was high. Synchrony also increased with cou-pling. Maximum synchrony occurred when coupling was high-est and noise was near 170 pA2 (Fig. 6C). This synergisticrelationship provides a means by which small networks canreach synchrony for a wide range of coupling and noise. In anetwork where either noise or coupling is low, increasing theother parameter will increase synchrony. Even baseline noise(12,000 �V2) or very small coupling (Cmax � 0.1) can bebrought to synchrony by this method. Conditions in which bothcoupling and noise increase simultaneously cause the greatestchange. This finding agrees with experimental evidence thatnoise is increased before an epileptiform burst (McBain andDingledine 1992) and also after stimulation (Manabe et al.1992; Mennerick and Zorumski 1995) and coupling increasesin some forms of epileptic activity (Dudek et al. 1986; Jefferys1994; Perez Velazquez and Carlen 2000). CR in this physio-logical model therefore provides an interesting insight into thepossible role of noise and coupling in the CNS: it is a simplemodel of epileptic bursting. These results suggest that epilep-tiform activity can be evoked in a network by increasingexcitability through a combination of increased noise and cou-pling that separately would not be sufficient to cause bursting.





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Network contributions to signal detection

The network was configured to test two main hypotheses ofsignal detection. First, the effects of coupling and the numberof cells on detection of a global signal were tested. With Cmax

� 0, the results are similar to the findings using the FitzHugh-Nagumo model (Collins et al. 1995) that showed the peakresponse increasing with N. However, the CA1 array did notrespond to a broad, nonspecific range of noise variance. This isbecause oscillations at high variance and the limited ability ofhigher amplitude minis (1 nS) to improve detection bothcaused low SNR. It is possible that higher amplitude minis andlarger N would produce the broad response described by Col-lins et al. (1995) and Moss and Pei (1995), but the key issue isthat physiological noise sources in CA1 cells cause oscillationsand will corrupt detection in a manner not predicted by SRtheory. When coupling increased, our results were only par-tially similar to previous coupled models for low noise levels—the peak resonance shifts up and to the left for increasingcoupling (Inchiosa and Bulsara 1995a)—but deviated fromthose predicted by SR at high noise. A coupled network cantherefore detect signals better than a single cell as long as noisedoes not induce periodic oscillations in the network. Theseresults suggest that neural networks in the brain are able toutilize noise and coupling to improve detection under certainconditions. First, a network with more cells can detect a sub-threshold signal with a higher SNR, leading to a strongeroutput. In the case of the CA1 cells, one functional conse-quence of this effect would be improved memory formation.Second, a highly coupled network is able to detect a signalfurther from threshold if noise is present. Functionally, such anetwork would be a sensitive detector. As suggested by Huberet al. (1999, 2000) for psychiatric illnesses, noise may play amajor role in brain pathology as well. In this case, the networkwould also be prone to oscillations if noise rose too high andcould become an epileptic focus. Obviously, these effectswould have strong implications in both normal and patholog-ical signal detection in the brain.

The second hypothesis was that a single cell could recruitother cells in the network through the combined effects ofcoupling and noise. With physiological noise sources, thesignal cell was unable to recruit its neighbors under mostcircumstances. In most cases, the excitability of the other cellsoverwhelmed the input signal with random firing. Even asuprathreshold signal had poor recruiting ability, suggestingthat recruitment in this network was difficult to produce usingthe chosen coupling mechanism regardless of signal strength.However, there were configurations in which the signal cellcould recruit the other cells: when noise and coupling werehigh enough to bring the network near the phase transition toperiodic firing. This is an interesting combination of the effectsof CR and SR in the network. In these situations, a networkprone to fire nearly synchronized bursts around a certain fre-quency could be driven to fire synchronously by a subthresholdperiodic signal at slightly higher frequency. This CA1 networkwas specifically designed to ignore frequency effects in thesignal by not including the potassium AHP channel or long-term potentiation (LTP) effects (Stacey and Durand 2000).However, this tuning phenomenon is frequency dependent andappears to be due to intrinsic membrane and network charac-teristics. This effect may provide a means for the network to be

tuned to detect and fire at a specific frequency. Physiologically,this effect could used by higher brain centers to tune CA1 orother coupled neurons to detect specific frequencies by merelychanging the variance of presynaptic firing.

C O N C L U S I O N

In summary, by using a CA1 model previously tested ex-perimentally, we were able to construct a network, determinethe effects of coupling and noise on signal detection, and relatethe results to physiological data. The network showed largeimprovements in SNR as coupling and N increased for noisenear baseline, similar to other work in AESR. At high noiselevels, the SNR did not match values predicted by SR due toperiodic oscillations. These oscillations, as well as the specificcharacteristics of the SNR, were strongly dependent on thequantal amplitude of the noise, showing the importance ofaddressing this parameter rather than merely the overall vari-ance. The highly coupled network was very sensitive to spe-cific driving signals and was able to synchronize to them. Thuswhile SR does appear to be feasible under physiological con-ditions in this network, there are interesting deviations from SRtheory that may have deep implications in the CNS. Theseresults demonstrate the importance of relating neural AESR tophysiological parameters to understand its possible role invivo. CR in this model demonstrated the synergistic role ofcoupling and noise in making the network synchronize, and isa simple model of epileptiform activity as well as being the firstdemonstration of coherence resonance in a vigorous physio-logical neural model. These studies demonstrate interestingphysiological implications of both coupling and noise on signaldetection in the brain, using the framework of CR and AESRas investigative tools.

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