Physics Letters A 377 (2013) 2585–2589

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Physics Letters A

www.elsevier.com/locate/pla

Inverse stochastic resonance induced by synaptic background activitywith unreliable synapses

Muhammet Uzuntarla ∗

Bulent Ecevit University, Engineering Faculty, Department of Biomedical Engineering, 67100 Zonguldak, Turkey

a r t i c l e i n f o a b s t r a c t

Article history:Received 11 May 2013Received in revised form 2 August 2013Accepted 4 August 2013Available online 12 August 2013Communicated by R. Wu

Keywords:NoiseUnreliable synapseNeural dynamicBifurcation

Inverse stochastic resonance (ISR) is a recently pronounced phenomenon that is the minimum occurrencein mean firing rate of a rhythmically firing neuron as noise level varies. Here, by using a realistic modelingapproach for the noise, we investigate the ISR with concrete biophysical mechanisms. It is shown thatmean firing rate of a single neuron subjected to synaptic bombardment exhibits a minimum as the spiketransmission probability varies. We also demonstrate that the occurrence of ISR strongly depends on thesynaptic input regime, where it is most prominent in the balanced state of excitatory and inhibitoryinputs.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

As is well known, neurons communicate with each otherthrough synapses and the process called synaptic transmission.Synaptic transmission is an essentially probabilistic process dueto the random neurotransmitter release of the synaptic vesicles.At some synapses the stochastic nature of synaptic communica-tion may give rise to highly unreliable transmission which hasbeen confirmed by well-designed biological experiments [1–8]. Forexample, in the cortex, it is found that the probability of neuro-transmitter release in response to a single spike can be as lowas 0.1 or lower, indicating that as many as 90% out of all arriv-ing presynaptic inputs fail to evoke a postsynaptic response [4].Moreover, some recent theoretical studies have also suggested thatthe transmission unreliability might be a part of the neural com-putation in the brain and possibly have significant implications ininformation processing [8–15].

On the other hand, in the past decades, noise in neurons hasattracted more and more attention due to its potential facilitatingeffects on information processing in nervous system. Researchersespecially paid close attention to stochastic resonance (SR) mecha-nism whereby generally feeble input information can be amplifiedand optimized by the assistance of noise [16–22]. More precisely,when the noise level is small, the neurons are not able to de-tect the signal due to its small amplitude, but as the noise raises,the temporal output becomes highly correlated with the signal

* Tel.: +90 372 257 4010; fax: +90 372 257 4023.E-mail address: muzuntarla@yahoo.com.

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resulting an increase in signal to noise ratio. Finally, for very largenoise intensities, the neuronal output is dominated by the noiseand the signal cannot be detected. Such an input–output rela-tionship of neurons exhibits a well-known bell-shaped structureas a function of noise intensity. Besides the noise, informationtransmission delays among neurons are crucial for the SR in neu-ral systems. In [23,24], it has been identified that the synaptictransmission delays lead to emergence of multiple stochastic res-onance peaks in networks of neurons, indicating that noise andinformation transmission delays can play complementary roles inwarranting optimal detection of weak signals. In contrast to SR,recent studies have concentrated on an inhibitory effect of noisein rhythmically firing neurons [25–30], that is, there exists a pro-nounced minimum in the firing rate as the noise level increases.Such an inhibition effect of noise has also been demonstrated ex-perimentally in squid giant axon operating as a pacemaker [31].Since the dependency of neuronal response on noise is reverseof that in SR mechanism, this new phenomenon is called inversestochastic resonance (ISR). In most previous modeling studies onISR, noise has been generally considered an external additive noisycurrent source appearing in the membrane potential equation, andassumed to be originated from the cumulative effect of overallnoise sources. However, such an approximation in noise model-ing is lack of mimicking the actual biophysical conditions and doesnot provide us a clear understanding the phenomenon with con-crete biological mechanisms.

Due to the large number of synaptic contacts in in vivo con-ditions, neurons are exposed to intense and random incomingexcitatory and inhibitory spike inputs. Based on the experimen-tal findings from electrophysiological studies [32–35], this synaptic

2586 M. Uzuntarla / Physics Letters A 377 (2013) 2585–2589

background activity is widely accepted to be the major source ofnoise in neurons. It is thus necessary to carry out further stud-ies by considering the realistic models of synaptic noise to clar-ify the underlying biophysical mechanisms which give rise to ISR.In this work, we investigate the ISR by using a detailed model-ing approach for the synaptic background activity with unreliablesynapses. We mainly examine influence of synaptic unreliability aswell as other important some background activity parameters, suchas the synaptic input regime, presynaptic firing rate and couplingstrength on the occurrence of ISR.

2. Mathematical model and setup

The system under study consists of a postsynaptic neuronwhich receives uncorrelated network activity from a finite numberof excitatory and inhibitory neurons. The time evolution of mem-brane potential of the postsynaptic neuron is modeled based onthe second-order Morris–Lecar (ML) equations as follows [36,37]:

Cdv

dt= −gCam∞(v)(v − V Ca) − gK w(v − V K) − gL(v − V L)

+ Iapp + Isyn(t) (1a)

dw

dt= φ

w∞(V ) − w

τw(V )(1b)

where v and w represent the membrane voltage and the activationof delayed-rectifier K+ current, respectively. C is the membranecapacity per unit area and φ is a constant that determines scal-ing rate for K+ channel opening. The parameters gx (x = Ca,K, L)

are the maximal conductance of calcium, potassium and leakagechannels, respectively. V Ca, V K and V L denote the correspondingequilibrium potentials. The parameters m∞ and w∞ stand for thefraction of open calcium and potassium channels at steady state,respectively; and they are given by the following equations:

m∞(V ) = 0.5

[1 + tanh

(v − V 1

V 2

)](2a)

w∞(V ) = 0.5

[1 + tanh

(v − V 3

V 4

)](2b)

with a time constant for the activation of potassium channels:

τw(V ) = 0.5

[cosh

(v − V 3

2V 4

)]−1

(2c)

where V 1 and V 3 are the activation midpoint potentials at whichthe corresponding currents are half activated; V 2 and V 4 denotethe slope factors of the activation. Finally, Iapp is an external cur-rent stimulus in μA/cm2. Notably, ML neuronal model can demon-strate two different types of neuronal excitability (i.e. class I andclass II excitability) when the model variables are set appropriately.Here, following the previous computational studies on ISR [25–30],we consider the class II excitability and set the ML model pa-rameters as: C = 20 μF/cm2, gL = 2.0 μS/cm2, gCa = 4.4 μS/cm2,gK = 8.0 μS/cm2, V K = −84 mV, V L = −60 mV, V Ca = 120 mV,V 1 = −1.2 mV, V 2 = 18 mV, V 3 = 2 mV, V 4 = 30 mV andφ = 0.04.

Finally, in Eq. (1a), Isyn is the total synaptic current introducedinto the neuron due to the network activity. We assume that themodel neuron receives a large number of excitatory and inhibitoryspike inputs from totally N presynaptic neurons. The ratio of ex-citatory to inhibitory synaptic contacts is taken as NE : Ni = 4 : 1to ensure physiological values found in in-vivo conditions [38,39].By considering the presynaptic neurons as a group that generatesindependent Poisson spike trains with the same input firing ratef in , the total synaptic current reaching the soma of the postsynap-tic neuron is given by:

Isyn(t) = wexc

[Ne∑

k=1

∑l

hlkδ

(t − tl

k

) − KNi∑

m=1

∑n

hnmδ

(t − tn

m

)](3)

where wexc represents the coupling strength for the excitatorysynapses and K is the relative strength between inhibitory andexcitatory synapses. tl

k is the discharge time of the l-th spike atthe excitatory presynaptic neuron k, and hl

k is the synaptic trans-mission reliability parameter of this spike which is used to mimicwhether the spike transmission is successful or not. Similarly, tn

m isthe discharge time of the n-th spike at the inhibitory presynapticneuron m, and hn

m is the synaptic transmission reliability parame-ter. The reliability of spike transmission is modeled based on thestochastic Bernoulli on–off process by assuming hl

k = 1 or hnm = 1

with probability ps , and hlk = 0 or hn

m = 0 with probability 1 − ps ,where ps is defined as the successful transmission probability ofspikes [10,18,40–42].

3. Results and discussions

We systematically analyze the ISR phenomenon in a single neu-ron subjected to a synaptic background activity, and discuss therelative contributions of different synaptic subunits on such a phe-nomenon. In the results presented below, the total number ofinput synapses is set N = 5000. Because it is a slow varying param-eter compared with other system parameters, we do not changeits value. Following the procedure in [25–28], the temporal out-put of the neuron is recorded for T = 20 000 ms (after eliminating1000 ms time interval as transient), and then the mean firing rateis calculated by counting the number of spikes and dividing it bythe recording interval T . For statistical accuracy, the entire pro-cedure is repeated 1000 times with random initial conditions of(v0, w0), and we finally compute the mean firing rate as the mea-sure of ISR.

As a starting point, we first consider the balanced state of exci-tatory and inhibitory synaptic inputs, and investigate whether theISR can be induced in this special regime by the synaptic back-ground activity with unreliable synapses. To do so, we set thesystem parameters as K = 4, wexc = 0.05, f in = 32 Hz, and com-pute the mean firing rate as a function of successful transmissionprobability ps for several typical values of the applied current Iapp .Fig. 1 features the obtained results. As seen in Fig. 1, with decreas-ing values of ps (right to left), all the mean firing rate curves dropat first until reaching some minimum and then begin to rise, ex-cept Iapp = 88, and finally attain to some saturated values. It is alsoseen that the minimum in mean firing rates occur at some corre-sponding optimal ps , indicating that appropriate tuning of synapticreliability can suppress the spiking activity even in the presenceof suprathreshold input current Iapp . This is a clear signature ofsynaptic transmission reliability induced ISR. Furthermore, it is ev-ident that ISR can significantly be modulated by Iapp: inhibitioneffect of synaptic transmission reliability on neuronal firing tendsto disappear as Iapp increases.

The underlying effects of ps and Iapp on the occurrence ofsynaptic reliability induced ISR can be understood by consider-ing the dynamical structure of the model neuron. The bifurcationdiagram in Fig. 2 shows that the deterministic ML model equa-tions may exhibit different solutions as Iapp varies. Namely, whenIapp < 88.29 = I1, there is only a stable equilibrium, generallyknown as “stable fixed point (SFP)”, corresponding to the restingstate of the membrane potential. At I1, a saddle node bifurcationgives rise to occurrence of stable and unstable limit cycles. Notably,the stable limit cycle (SLC) corresponds to regular spiking behaviorof the neuron. With further increase in Iapp , although the SLC doesnot change very much, the unstable limit cycle (ULC) collapsesonto the SFP at I2 = 93.86 through a subcritical Hopf bifurcation.

M. Uzuntarla / Physics Letters A 377 (2013) 2585–2589 2587

Fig. 1. Inverse stochastic resonance induced by the synaptic transmission reliabil-ity. Figure shows the mean firing rate of ML neuron versus synaptic transmissionprobability for different values of Iapp . Evidently, the output of the system quanti-fied by the mean firing rate exhibits a reversed bell-shaped dependence on ps formost of the applied current values of interest. The employed system parameters are:wexc = 0.05, K = 4, f in = 32 Hz.

Fig. 2. Bifurcation diagram of the Morris–Lecar neuron for the parameter, Iapp . Thesolid line shows the stable fixed point (SFP) for Iapp < I1. The dashed line standsfor the unstable fixed point (UFP) above I2. Extreme values of periodic solutionsare shown with dots and unfilled-circles illustrating the stable limit cycle (SLC) andunstable limit cycle (ULC) attractors, respectively. The vertical dashed lines at I1

and I2 divide the Iapp parameter interval into three regions where the ML neuronexhibits different phase portraits in each one.

Also, this second bifurcation destroys the stability of the fixedpoint. Between these two bifurcation points, i.e., I1 < Iapp < I2, SFPand SLC solutions coexist and an ULC separates their respective at-traction basins. Thus, in this region of Iapp , the membrane potentialof the deterministic ML neuron undergoes either the subthresholdresponse or the regular spiking response depending on the initialconditions (v0, w0).

With this understanding, we now return to Fig. 1 to discuss themechanism that gives rise to ISR for different values of Iapp . Forthe case of Iapp < I1 (see Iapp = 88 curve), since there is only aSFP as an attractor, membrane potential cycles around this pointon a noisy orbit due to the synaptic background activity. Whenthe spike transmission reliability is high, uncorrelated presynap-tic spike trains cause the generation of a fluctuating synaptic noisy

current. At some instants of time, this highly fluctuating synap-tic current introduced into the neuron leads to the membranepotential trajectory cross boundary of deterministic dynamics re-sults in wider circuits of cycling before spiraling into the equi-librium. Frequency of these sudden crossings and back again tothe SFP depends on the intensity of synaptic current fluctuationswhich is determined by the incoming spikes from presynaptic neu-rons. Therefore, probability of spike transmission is the key fac-tor that controls the synaptic noise strength. With the decreasingvalues of ps , since many of the presynaptic spike inputs fail toevoke postsynaptic currents, the total synaptic current is no longerenough to move away the trajectory from the SFP. Consequently,the mean firing rate curve converges to zero and stays there in-dicating that the ISR cannot occur for the values of Iapp below I1for any spike transmission probability. On the other hand, for thecase of I1 < Iapp < I2 shown in Fig. 1, there are two main attrac-tors: one is SFP and the other is SLC. In this case, synaptic inputswith high transmission reliability, which result in large fluctua-tions in synaptic current, can drive the membrane potential of theML neuron from one stable solution to the other. The frequencyof random switchings between these stable states decreases withthe decrease in spike transmission probability. Consequently, themean firing rate curves reach their minimums for moderate valuesof ps . Why, then, does the mean firing rate increase again as ps de-creases further? This can be attributed to the dynamical structureof the ML neuron model where the ULC acts as an attractor basinboundary for the transitions between SLC and SFP. Based on themathematical theory given in [43], the switching region betweenSLC and ULC is smaller than that of the region between ULC andSFP. For small ps , transitions from SLC to SFP are rare and needmore time due to the low level of perturbations in synaptic cur-rent, and therefore the persistently spiking trajectories contributemany spikes to the mean firing rate. Thus, the ISR curves rise withthe decreased values of spike transmission probability ps .

In earlier works on ISR [25–30], the phenomenon was estab-lished via varying the strength of the ideal additive Gaussian whiteor colored noise, while keeping mean value at zero. When therelative coupling strength factor between inhibitory and excita-tory synapses is K = 4 in the model studied here, the ML neu-ron receives balanced synaptic inputs that result in a fluctuatingsynaptic current with zero mean value. Here, when other sys-tem parameters are kept constant, we have observed the samebehavior of firing rate curves by varying ps suggesting that thesynaptic transmission reliability might be an important factor tocontrol the strength of the synaptic current fluctuations. Accord-ing to the previous theoretical results based on the mean fieldanalysis [18,44,45], its actual impact on the intensity of synap-tic noise arises from determining the effective mean spike arrivalrate per synapse by tuning the input firing rate f in . To illustratethis, we further obtain the mean firing rates in (ps, f in) domainfor a fixed input current Iapp = 89 where the ISR is most promi-nent. Fig. 3 features the obtained results. As we see, there existsan optimal (ps, f in) region where the mean firing rate of the neu-ron is minimal (black area in the figure). Therefore, it is now clearthat ISR occurs in our model neuron by the combined effect ofthese two parameters. Notably, in the figure, each green “+” andwhite “∗” marks refer to the minimums in average firing rates forthe corresponding ps and f in , respectively. These data points mightthemselves be well fitted by the line ps fin ≈ 1 Hz, indicating thatif and only if the effective mean spike arrival at such critical valueis able to provide required stochastic fluctuations for the occur-rence of ISR according to our above discussions.

We next investigate the impact of excitatory coupling strength(wexc) on the unreliable synaptic transmission induced ISR.To do so, we compute the mean firing rates as a function of ps

for different values of wexc by a fixed value of applied current

2588 M. Uzuntarla / Physics Letters A 377 (2013) 2585–2589

Fig. 3. (Color online.) Mean firing rate of the ML neuron as functions of the spiketransmission probability ps and the presynaptic input firing rate f in in the balancedstate of excitatory and inhibitory inputs. In the figure, each green “+” and white“∗” marks refer to the minimums in mean firing rates for the corresponding ps

and f in , respectively. All these minimum data points can be well fitted by the lineps fin ≈ 1 Hz (blue line). The employed system parameters are wexc = 0.05, K = 4,Iapp = 89.

Fig. 4. Mean firing rate of the neuron in dependence on spike transmission prob-ability ps for different values of excitatory coupling strength wexc . The employedsystem parameters are: Iapp = 89, K = 4, f in = 32 Hz.

Iapp = 89. Fig. 4 presents the obtained results. It is seen that de-creasing wexc does not influence the minimums in mean firingrates, and only shifts the corresponding optimal ps value to rightindicating that more reliable synapses are needed for the occur-rence of ISR as the excitatory coupling strength decreases. This isno of surprise, since under this condition (balanced state of synap-tic inputs) the mean value of the total synaptic current is zero,and therefore, changing wexc only affects the fluctuations in to-tal synaptic current. When the excitatory coupling is strong, eachpresynaptic spike input with a high transmission probability re-sults in large deviations in synaptic current. In addition to thateffect of wexc on synaptic fluctuations, there should be a properintensity of noise for the occurrence of ISR as discussed above.Considering that the two mechanisms together, it is easy to under-stand that why the ISR curve shifts to lower values of ps as wexc

increases.

Fig. 5. Mean firing rate versus spike transmission probability for different presynap-tic input regimes. A lower and higher value of K than the value of balanced state(K = 4) indicates the high and low input regime, respectively. The employed pa-rameters are: wexc = 0.05, f in = 32, Iapp = 89.

So far, we have examined the ISR phenomenon in the balancedstate of excitatory and inhibitory synaptic inputs by setting thescale factor K = 4. To investigate how the synaptic input regimeaffects the ISR, we again computed the mean firing rates as a func-tion of spike transmission probability for different values of the K ,by a fixed value of input current Iapp = 89 and excitatory cou-pling strength wexc = 0.05. Obtained results are presented in Fig. 5.We find that the spike transmission reliability induced ISR dependsstrongly on the relative coupling strength between inhibitory andexcitatory synapses. As K decreases, where the neuron operatesin the high input regime, the amplitude of the inverse resonancepeaks first weakens (see K = 3) and disappears completely withfurther decrease in K . On the other hand, in the case of low in-put regime (K > 4), we still cannot observe ISR phenomenon asps varies with this setup of system parameters. In this regimeof synaptic inputs, average firing rates get zero values up to cer-tain ps , and then start to increase with further decrease in ps

until reaching to a saturated firing rate value. Such an effect canbe understood by considering the significant influence of K in de-termining the sign of the total synaptic current introduced intothe neuron. More precisely, when K = 4, since the excitatory andinhibitory inputs cancel each other out, average synaptic currenthas zero mean value. The spike transmission probability ps doesnot have any influence on average synaptic current in the balancedstate, but it directly controls the intensity of fluctuations. For thehigh input regime (K < 4), the contribution of excitatory inputsto the total synaptic current is more than that of inhibitory ones.Thus a smaller K introduces a positive average synaptic currentinto the neuron dynamics for a fixed ps . Also, it is worth nothingthat increasing ps in the high input regime amplifies the averageand fluctuations of total synaptic current positively. A positive signsynaptic current together with Iapp move away the neuron fromthe bifurcation point at I1 to I2 or further where the membranepotential trajectory most probably stays in SLC state (persistentlyspiking behavior). Thus, in this regime, moderate values of ps arenot able to provide necessary synaptic current fluctuations to trig-ger the above-mentioned mechanism that gives rise to ISR. On theother hand, when K > 4, since the inhibitory inputs dominatethe excitatory ones, the neuron receives a negative average synap-tic current where the intensity of its fluctuations is controlled bythe spike transmission probability, i.e. increasing ps negatively and

M. Uzuntarla / Physics Letters A 377 (2013) 2585–2589 2589

positively amplifies the average and fluctuations of synaptic cur-rent, respectively. For large values of ps , the total external inputcurrent (Isyn + Iapp) introduced into the neuron becomes highlysubthreshold (below I1) where the membrane trajectory of theneuron stays most probably at SFP basin. Although the intensityof fluctuations is very high for large values of ps , it is not suffi-cient to drive the membrane potential trajectory from SFP to fireaction potentials. Therefore, the mean firing rate gets small val-ues in the low input regime for large ps (see K > 4 curves inFig. 5). Finally, since both the negative and positive amplificationsin synaptic current statistics weaken as ps → 0, the net externalcurrent input to the neuron approaches to Iapp with very low fluc-tuations, and accordingly the membrane trajectory of the neuronmost probably operates in SLC attractor basin. This is the reason ofincrease in mean firing rate for small values of ps in the low inputregime.

4. Summary

A challenging research issue in neuroscience is the understand-ing the influence of noise in neural dynamics. In this context, in-verse stochastic resonance is a recently pronounced phenomenonthat is the minimum occurrence in the mean firing rate of a rhyth-mically firing neuron as noise level varies. In most studied cases ofISR, noise is generally considered an ideal additive Gaussian whitenoise. Although considering the neuronal noise as the Gaussianwhite noise is in favor of both the modeling analysis and fast com-putation, it does not provide a realistic approach to understand thephenomenon with concrete biological mechanism. Therefore, it ishighly relevant to investigate which features of actual neural sys-tem may give rise to occurrence of ISR.

In this Letter, we have studied the ISR in a single ML neuronreceiving a large number of stochastic excitatory and inhibitoryspike inputs through unreliable synapses. By introducing a stochas-tic on–off process to model the unreliable synaptic transmission,we systematically investigated how the synaptic unreliability andother some synaptic parameters influence the ISR phenomenon.First, we have showed that the key factor that gives rise to ISRis the dynamical structure of the model neuron where the stableattractors (SFP and SLC) are in close proximity to each other. Thisproximity indicates that the model neuron is highly sensitive tonoisy fluctuations of membrane potential which move the trajec-tory from one stable state to the other. In our recent work [46], wehave explained this dynamical picture in detail. As the main goalof the present study, we showed that synaptic unreliability caninduce ISR behavior in our model neuron by decreasing or evensuppressing the spiking activity at some particular values of spiketransmission probability. To the best our knowledge, this is the firststudy demonstrating that the ISR occurs in neural systems as aconsequence of unreliable synaptic transmission. It is also shownthat such a role of unreliable synapses is due to their direct effectin determining the effective mean spike arrival rate per synapse(see Fig. 3). On the other hand, we also examined the influenceof excitatory coupling strength and the synaptic input regime onthe unreliable synaptic transmission induced ISR. Under certain

conditions, our simulations have revealed that the ISR behaviormay be modulated by these two important background activity pa-rameters. In particular, the increased values of excitatory couplingshift the required synaptic transmission probability to the lowervalues for the appearance of minimum in mean firing rate. Besidesthis, we have also observed that the ISR behavior strongly dependson the synaptic input regime, where it is most prominent in thebalanced state of excitatory and inhibitory inputs, and it does notoccur in low input regime.

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