Signal propagation through feedforward neuronal networks with different operational modes

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1 OFFPRINT Signal propagation through feedforward neuronal networks with different operational modes Jie Li, Feng Liu, Ding Xu and Wei Wang EPL, 85 (2009) Please visit the new website

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3 February2009 EPL, 85(2009)38006 doi: / /85/ Signal propagation through feedforward neuronal networks with different operational modes JieLi,FengLiu (a),dingxuandweiwang Nanjing National Laboratory of Microstructures and Department of Physics, Nanjing University Nanjing , China received 30 August 2008; accepted in final form 19 January 2009 publishedonline13february2009 PACS Sn Neural networks and synaptic communication PACS Xt Synchronization; coupled oscillators PACS L- Neuroscience Abstract How neuronal activity is propagated across multiple layers of neurons is a fundamental issue in neuroscience. Using numerical simulations, we explored how the operational mode of neurons coincidence detector or temporal integrator could affect the propagation of rate signals through a 10-layer feedforward network with sparse connectivity. Our study was based on two kinds of neuron models. The Hodgkin-Huxley(HH) neuron can function as a coincidence detector, while the leaky integrate-and-fire(lif) neuron can act as a temporal integrator. When white noise is afferent to the input layer, rate signals can be stably propagated through both networks, while neurons in deeper layers fire synchronously in the absence of background noise; but the underlying mechanism for the development of synchrony is different. When an aperiodic signal is presented, the network of HH neurons can represent the temporal structure of the signal in firing rate. Meanwhile, synchrony is well developed and is resistant to background noise. In contrast,ratesignalsaresomewhatdistortedduringthepropagationthroughthenetworkoflif neurons, and only weak synchrony occurs in deeper layers. That is, coincidence detectors have a performance advantage over temporal integrators in propagating rate signals. Therefore, given weak synaptic conductance and sparse connectivity between layers in both networks, synchrony does greatly subserve the propagation of rate signals with fidelity, and coincidence detection could beofconsiderablefunctionalsignificanceincorticalprocessing. Copyright c EPLA, 2009 Introduction. The essence of cortical functions is the propagation and transformation of neuronal activity by cortical circuits. Theoretical analyses of signal propagation have mainly focused on models of feedforward networks composed of layers of neurons. How information is coded within such networks has been hotly debated[1 3].Itcanbecarriedeitherbytheratesatwhich neurons discharge spikes[1] or by precise spike timing[2]. A novel phenomenon synchrony-based propagation of rate signals through multiple layers has recently been reported both experimentally[4] and theoretically[5 8]. It is shown that when constant-frequency signals are delivered to the network, neuronal firing is asynchronous for the first three layers but becomes progressively more synchronous in successive layers. Neurons in deeper layers fire synchronously, and rate signals cannot be stably propagated without synchrony. Under the condition of the (a) fliu@nju.edu.cn experiment, however, rate signals are somewhat distorted during the propagation of time-varying inputs [4]; the effect of background noise on signal propagation has not been investigated in detail[5]. Two issues naturally arise, i.e., what biophysical mechanism underlies the robustness of synchrony and under what conditions temporally changing inputs can be encoded by feedforward networks. Here, we propose that the operational mode of neurons is closely related to signal propagation. Generally, cortical neurons can operate in two distinct ways, behaving as coincidence detectors or temporal integrators[9]. In the present work, we construct a 10-layer feedforward network composed of spiking neurons with sparse connectivity.neuronscanoperateinoneoftwomodesviavarious mechanisms. For simplicity, here we use the Hodgkin- Huxley(HH) and the leaky integrate-and-fire(lif) model to represent coincidence detector and temporal integrator, respectively. However, it is worth noting that only the HH and LIF neurons with suitable parameters(as chosen in p1

4 JieLietal. Fig. 1: Propagation of rate signals through two types of feedforward networks in the absence of background noise. White noise is afferent to the input layer.(a) Schematic illustration of a 10-layer feedforward network, with each layer composed of 200 neurons.eachneuronreceivesabout20synapticinputsfromtheprecedinglayer.(b)spatiotemporalfiringpatternsfordifferent layersofthehhfn(left)andliffn(right).(c)thecoherencemeasurek iand(d)themeanfiringratef ivs.layerindex for the HHFN( ) and LIFFN(+), respectively.(e) Firing activity of the 10th neuron in layer 2. Left: coincidence detection operatingonatrainofpscs(lowertrace)bythehhneuron.right:temporalintegrationofatrainofpscsbythelifneuron. The upper traces show the time course of the membrane potential. this work) can act as the coincidence detector and integrator, respectively. Moreover, it is well known that the two models operate quite differently[10]. Nevertheless, our aim here is not to compare their distinct dynamic behaviors but to explore how the operational mode influences signal propagation. The networks composed of HH and LIF neurons are called HHFN and LIFFN, respectively. Our results show that the operational mode of neurons affects the emergence and robustness of synchrony as well as thefidelityofratecoding.thehhfnhasaperformance advantage over the LIFFN. In the presence of background noise, the HHFN can faithfully propagate time-varying inputs with synchrony well developed in deeper layers. The concept of coincidence detection thus extends the synchrony-based rate coding since it allows for the representation of the temporal structure of input signals. Model. The architecture of the 10-layer feedforward network is illustrated in fig. 1(a). Each layer is composed of 200 neurons, and each neuron randomly receives synaptic inputs from 10% of neurons in the preceding layer. There are no couplings among neurons within the same layer p2

5 Signal propagation through feedforward neuronal networks with different operational modes The dynamic equations for HH neurons are written as follows: C m dv i,j dt = g Na m 3 i,jh i,j (V i,j E Na ) g K n 4 i,j(v i,j E K ) g l (V i,j E l ) I syn i,j (t)+η i,j(t)+i 0 i+δ i,1 s(t) togetherwithdm i,j /dt=α m (V i,j )(1 m i,j ) β m (V i,j ) m i,j,dh i,j /dt=α h (V i,j )(1 h i,j ) β h (V i,j )h i,j,and dn i,j /dt=α n (V i,j )(1 n i,j ) β n (V i,j )n i,j.v i,j denotesthe membrane potential of the j-th neuron in layer i.the same functions and parameters are used as in refs.[5,11]. Especially,C m =1µF/cm 2 andg l =0.3mS/cm 2.Thatis, the passive membrane time constant is 3.3 ms. For the LIF neuron, a spike is discharged each time membranepotentialv reachesafiringthresholdv th.v is then reset to V reset and stays there for an absolute refractory period τ ref. Below threshold, V obeys the following equation[12]: dv i,j C m = g l (V i,j E l ) I syn i,j dt (t)+η i,j(t)+ii+δ 0 i,1 s(t). The parameter valuesareasfollows: C m =0.2nF,g l = 0.01µS, E l = 70mV, V reset = 60mV, V th = 50mV, and τ ref =2ms. Thus, the passive membrane time constantis20ms. Inbothnetworks,I 0 i isaconstantbias,andasignal s(t)isdeliveredonlytolayer1.s(t)obeysthefollowing equation: ds dt = s τ s + g w(t) τ s, whereτ s isthecorrelationtime,andg w (t)isthegaussian white noise with g w (t) =0 and g w (t 1 )g w (t 2 ) = 2D g δ(t 1 t 2 ).τ s issetto50msforbothnetworks;d g is500µa 2 ms/cm 4 forthehhfnand5.2na 2 msforthe LIFFN. The background noise is assumed to be Gaussian whiteandindependentofanyother,i.e., η i,j (t) =0and η i,j (t 1 )η i,m (t 2 ) =2D i δ j,m δ(t 1 t 2 ).D i isreferredtoas the noise intensity of layer i. For simplicity, we assume D 2 =D 3 = =D 10 =D s. Ni,j k=1 ThesynapticinputisdescribedasI syn i,j (t)=n 1 i,j g syn α(t t i 1,k )(V i,j (t) E syn )withα(t t i 1,k ) α(t )=(t /τ)e t /τ for t >0 and 0 otherwise. N i,j is thenumberofneuronsinlayeri 1coupledtothej-th neuron in layer i. t i 1,k is the firing time of the k-th neuron in layer i 1. The synaptic time constant is taken asτ=3mstomodelsuchfastsynapticcurrents as mediated by AMPA receptors. The synaptic reversal potential E syn is set to 0mV, implying that all the couplings are excitatory. Themeanfiringratef i iscalculatedbyaveragingon all neurons in layer i and over a time window of 2s (see fig. 1(d)), while the simultaneous firing rate r i is calculated by averaging over a time window of 40ms (see fig. 3). The degree of synchrony among neurons canbecharacterizedbyacoherencemeasurek i,which isanaverageofnormalizedcross-correlationk XY over allneuronalpairsinlayeri [5]. To compute K XY,a time interval T (T=2sforfig.1(c)orT=40msfor figs. 2(e) and 4(a)) is divided into k bins of γ=1ms, and two spike trains are given by X(l)=0or1and Y(l)=0or1(0and1correspondingtozeroandonespike, respectively),withl=1,2,,k(t/k=γ).thus,wehave K XY (γ)= k l=1 X(l)Y(l)/[ k l=1 X(l) k l=1 Y(l)]1/2.The integration method used to simulate the dynamics is a second-order stochastic algorithm[13], with a time step of 0.02 ms. Results. Wefirstconsiderthecaseinwhichwhite noiseispresentonlyinlayer1,i.e.,d 1 0andD s =0, intheabsenceofinputsignal.thisistomodelthecase whereneuronsinlayer1fireatconstantrates.tomake behaviors in both networks comparable, we keep their correspondingf 1 andf 10 identical,respectively.tothis end,weassumeg syn =0.62mS/cm 2,D 1 =3µA 2 ms/cm 4 and I 0 i =1µA/cm2 for the HHFN, and g syn =0.06µS, D 1 =0.6nA 2 msandi 0 i =0fortheLIFFN. In both networks, each neuron in layer 1 fires irregularly in response to white noise, and the spatiotemporal firing pattern exhibits a uniform distribution (fig. 1(b)). For layers2and3,thedotsinthefiringpatternsbeginto cluster, implying that neurons tend to fire synchronously. The clustering becomes progressively sharper in successive layers, and synchrony is well developed by layer 6. This tendency is prominent in both networks. Nevertheless, the HH neurons in deeper layers fire tonically, while the LIF neuronsfireinbursts. Thedegreeofsynchronycanbequantifiedbythe coherencemeasurek i.forbothnetworks,k i increases sigmoidally with layer and is saturated to 1(fig. 1(c)). Note that this tendency is persistent across a range of noise intensities(data not shown). On the other hand, themeanfiringratef i firstdecreasesmarkedlyandthen increases in both networks(fig. 1(d)). For deeper layers, however,f i becomessaturatedinthehhfn,whereasf i rises monotonically with layer in the LIFFN. This results from their distinct biophysical properties. Note that the saturation of firing rate in deeper layers has also been reported in refs.[4,5]. A key reason for synchrony in such feedforward networks isthatneuronsinanygivenlayersharealargequantity of common synaptic inputs. Here the connection probability between neighboring layers is 10%, and neurons share about 1% of the same synaptic inputs. This common input tends to evoke spikes within a restricted time window, leading to partial synchrony between corresponding postsynaptic neurons. The larger the connection probability, themorerapidlysynchronyisbuiltup.ontheotherhand, the operational mode of neurons plays a significant role. To demonstrate the distinct integration modes of two types of neurons, fig. 1(e) depicts the traces of the membrane potential and the corresponding postsynaptic currents(pscs) of some neuron in layer 2. Clearly, the p3

6 JieLietal. Fig. 2: Propagation of a time-varying signal in the presence of background noise.(a) Time course of an aperiodic signal.(b) Time courseofmembranepotentialsofneuronsinlayer1ofthehhfn(left)andliffn(right)intheabsence(uppertrace)or presence(lower two traces) of noise.(c) Spatiotemporal firing patterns for different layers of the HHFN(left) and LIFFN(right). (d)thehistogramofspikesdischargedbylifneuronsinlayer10withthetimebinbeing1ms.theinsetshowsthenumberof spikesdischargedbyeachlifneuronduringthesameperiod.(e)timecourseofk 10fortheHHFN(solidline)andLFFNN (dottedline).theparametervaluesareasfollows:d 1=1.1µA 2 ms/cm 4 and{i 0 i,i=1 10}={0, 2.1, 2.1, 2.2, 2.2, 2.3, 2.6, 2.6, 2.8, 2.9}(inunitofµA/cm 2 )forthehhfn,andd 1=0.2nA 2 msand{i 0 i}={0, 0.07, 0.07, 0.07, 0.07, 0.07, 0.08, 0.08, 0.08, 0.09}(inunitofnA)fortheLIFFN. presynaptic spikes in the first layer are dispersed in time. For the HH neuron, most postsynaptic currents do not actually contribute to the generation of spikes and only result in small fluctuations of membrane potential. In fact, only coincident synaptic inputs can effectively trigger postsynaptic spikes; that is, the HH neuron is most sensitive to presynaptic pulses arriving simultaneously, acting as a detector for the temporal coincidence of presynapticpulses.forthelifneuron,however,most, if not all, incoming PSCs contribute to the generation of spikes, for the membrane potential rises persistently untilaspikeistriggered.thus,thelifneuronactsasa temporal integrator. Combining the above discussions, we can interpret distinctfiringpatternsinthetwonetworks.inthehhfn, sincetheneuronsinlayer1firerandomly,thoseinlayers2 and 3 tend to fire only when sufficient numbers of presynaptic pulses arrive simultaneously, which leads toadecreaseinfiringrate.insuccessivelayers,onthe one hand, sparse firings cannot propagate across deep layers. On the other hand, each passing layer recruits more neurons to fire simultaneously. Thus, synchrony becomes more precise, while the firing rate increases until saturation occurs in deeper layers. In the LIFFN, since spikes discharged by neurons in layer 1 are uniformly distributed, neurons in layer 2 summate synaptic inputs over extended time intervals to fire. As the mean synaptic input is nearly the same, some neurons begin to fire simultaneously. Owing to small synaptic conductance, only large current transients can evoke downstream spikes. Neuronal firing becomes more synchronous in successive layers. Meanwhile, neurons in deeper layers can p4

7 Signal propagation through feedforward neuronal networks with different operational modes Fig.3:Timecourseofthesimultaneousfiringratesr 1andr 10 for the HHFN(a) and LIFFN(b), respectively. The results are averaged over 50 trials with different noise realization. The sameparametersasinfig.2. repetitively integrate synchronous synaptic inputs in the nearpastsothattheycanfireinbursts.thus,thefiring rate rises persistently with layer. Therefore, in the presence of sparse connectivity between neighboring layers and weak synaptic conductance, rate signals cannot be stably propagated through multilayer networks without synchrony. It is the combination of the network structure with the operational mode of neurons that determines the way in which rate signals are propagated. Moreover, the operational mode of neurons can remarkably affect the robustness of synchrony to noise, as shown in the following. Nowwetakeintoaccountbackgroundnoiseineach layer and keep D 1 =D s.thevaluesofd 1 and I 0 i are adjustedsothatthespontaneousfiringrateisabout5hz throughout both networks. Here the aperiodic signal s(t), showninfig.2(a),isdeliveredtolayer1.notethatthetwo networks respond quite differently to the presence of noise. Forexample,theHHneuronsinlayer1showamoderate distortion of spike trains after noise is presented, whereas the LIF neurons display totally distinct spike sequences (fig.2(b)). IntheHHFN,sinceallneuronsinlayer1receivethe same signal, they tend to fire synchronously despite noise, and the dots in the spatiotemporal firing pattern form the columns(fig. 2(c)). Moreover, these firings are temporally modulated by the signal. This occurs because the HH neurons can filter out background noise as coincidence detectors. In subsequent layers, for activity to propagate from one layer to the next, there must be sufficient synchronypresentintheinputreceivedbyneuronsinthe next layer. As a result, synchrony is well developed by layer4.indeeperlayers,noisehasaminoreffectonsignal propagation, only evoking small jitters in spike timing and additional few spikes. Overall, synchrony is robust to noise. IntheLIFFN,neuronsinlayer1integrateallinputs including the signal and noise and fire nearly randomly. Fig. 4: Effect of background noise on signal propagation. (a)k 10and(b)themaximumPearsoncoefficientofcorrelation betweenr 1andr 10vs.spontaneousfiringratefortheHHFN ( ) and LIFFN ( ), respectively. The results are averaged over 50 trials with different noise realization. The values of D 1andI 0 i areadjustedsothatthespontaneousfiringrateis f sthroughoutbothnetworks. NotethatI 0 i becomesmorenegativeinsuccessivelayers andthatnoisehasamarkedeffectonneuralfiring.only sufficiently large current transients can effectively trigger spikes in postsynaptic neurons. Neurons in deeper layers exhibit weak synchrony and fire in bursts, as seen in fig.2(d).duringthetimeintervalbetween600and800ms, for example, the numbers of spikes discharged by each neuron in layer 10 are comparable, and the spikes are nearly uniformly distributed within the interval. This is alsomanifestinfig.2(e),wherek 10 fortheliffnremains a very small value, whereas that for the HHFN fluctuates around 0.8. Therefore, although signals can be propagated through both networks, they exhibit different fidelity and robustness to noise. Sincetheinputsignalvarieswithtime,wehavetocalculate the simultaneous firing rate of each layer. Figure 3(a) showsthetimecourseofr 1 andr 10 forthehhfn.clearly, r 10 followsr 1 withatimelag,andtheyexhibitastrong temporal correlation. The maximum Pearson coefficient (P)ofcorrelationbetweenthemis0.76,whereasitisonly 0.22fortheLIFFN,whereonlytheoverallchangeofr 10 is consistentwiththatofr 1 (fig.3(b)).thus,thehhfnis capable of propagating both constant and time-varying rate signals, whereas the LIFFN may be better suited for conveying constant or slowly modulated rate signals. This also suggests that synchrony subserves the encoding of temporal structures of input signals across layers. Note that ref. [14] reported that cortical neurons in a twolayerednetworkcanfiresynchronouslyinthepresenceof weak noise. To further explore the effect of background noise on signal propagation, we systematically change the strength of noise intensity, which leads to different spontaneous firing rate f s. K 10 decreases gradually with increasing f s (fig. 4(a)). Over the whole range shown, however, K 10 for the HHFN exhibits a relatively large value, indicating strong synchronous firings in deeper layers. In contrast,k 10 fortheliffnisverysmall.thisfurther demonstrates that synchrony in the HHFN is robust to p5

8 JieLietal. noise. Moreover, the HHFN provides a good substrate for conveying temporal patterns in the signal, as seen infig.4(b)wherep isalwayslargerthan0.65.butthe LIFFN performs poorly in propagating rate signals since P is only around 0.2. This confirms that synchronous firings of neurons ensure the propagation of rate signals with fidelity. Discussion. In the present work, we have explored the propagation of rate signals in two types of feedforward networks. In the absence of background noise, rate coding can be realized with the help of synchrony in both networks,buttheunderlyingmechanismforthebuildup of synchrony is distinct. In the presence of noise, the HHFN can faithfully propagate the time-varying signal. Synchrony is well established in deeper layers and is resistant to noise. In contrast, the rate signal is somewhat distortedduringthepropagationthroughtheliffn,and neurons in deeper layers display weak synchrony. Thus, coincidence detectors have a performance advantage over temporal integrators in conveying rate signals through feedforwardnetworks. It has been demonstrated that synchronous firing of neuronscanplayawiderangeofrolesinbrainfunctions such as binding distributed features into a coherent representation[15]. Here we show that synchrony subserves the propagation of rate signals, which is more prominent in the presence of background noise. It ensures that repetitive current transients in successive layers are large enough to triggerspikesindownstreamneurons.suchafiringmode is characteristic of fast computation in cortex. That is, at any given time neurons that fire simultaneously may form a functional group representing a specific input feature, and this ensemble can be modulated very dynamically. Even the pattern of synchronization can flexibly determine the patterns of neuronal interactions, which may contribute to cognitive functions[16]. Our results suggest that coincidence detection is responsible for the robustness of synchrony. Detectors are more sensitive to synchronized inputs and are more easily evoked to fire synchronously. This is consistent with the results in refs. [17 19]. Moreover, neurons operating as coincidence detectors allow for much richer dynamics and can convey more information about temporal patterns in the input. This is consistent with a theoretical study based on abstract coincidence detectors[20]. Ontheotherhand,thereareafewmechanismsthat can make cortex neurons act as coincidence detectors. For example, the nonlinear interaction between spike frequency adaptation and increased neuronal membrane conductance leads to a switch from temporal integration to coincidence detection in pyramidal neurons[21]. Thus, coincidence detection might be a prevalent operational mode of cortical neurons and be of considerable functional significance in cortical processing[9]. Moreover, it is interesting to explore the effect of operational mode of neurons on signal propagation with the same neuron model. Conclusion. In conclusion, both constant and timevarying rate signals can be stably propagated through a sparsely connected feedforward network, provided that the component neurons operate as coincidence detectors. Synchrony greatly subserves the propagation of rate signals with fidelity. Finally, it is worth noting that cortical circuits are greatly endowed with collateral connections. Taking that into account will further clarify the underlying mechanisms for signal propagation. We thank the two anonymous referees for their helpful comments and suggestions. This work was supported by the NNSF of China ( ), the National Basic Research Program of China(2007CB814806) and the SRF forrocs,sem. REFERENCES [1] vanrossumm.c.w.,turrigianog.g.andnelson S.B.,J.Neurosci.,22(2002)1956. [2] Diesmann M., Gewaltig M. O. and Aertsen A., Nature,402(1999)529. [3] Litvak V., Sompolinsky H., Segev I. andabeles M., J.Neurosci.,23(2003)3006. [4] ReyesA.D.,Nat.Neurosci.,6(2003)593. [5] WangS.,WangW.andLiuF.,Phys.Rev.Lett.,96 (2006) [6] HamaguchiK.andAiharaK.,Neurocomputing,58-60 (2004)85. [7] CateauH.andReyesA.D.,Phys.Rev.Lett.,96(2006) [8] DoironB.,RinzelJ.andReyesA.,Phys.Rev.E,74 (2006) [9] König P., Engel A. K. and Singer W., Trends Neurosci., 19(1996) 130. [10] Feng J. and Zhang P., Phys. Rev. E, 63 (2001) [11] HanselD.,MatoG.andMeunierC.,Europhys.Lett., 23(1993)367. [12] Dayan P. andabbott L. F., Theoretical Neuroscience (MIT Press, Cambridge) 2001, p [13] FoxR.F.,Phys.Rev.A,43(1991)2649. [14] MasudaN.andAiharaK.,Phys.Rev.Lett.,88(2002) [15] SingerW.andGrayC.M.,Annu.Rev.Neurosci.,18 (1995)555. [16] WomelsdorfT.etal.,Science,316(2007)1609. [17] Galán R. F., Ermentrout G. B.andUrban N. N., Phys.Rev.E,76(2007) [18] Tateno T.andRobinson H.P.,J.Neurophysiol.,95 (2006)2650. [19] TatenoT.andRobinsonH.P.,Biophys.J.,92(2007) 683. [20] MikulaS.andNieburE.,NeuralComput.,17(2005) 881. [21] Prescott S. A., Ratté S., Koninck Y. D. and SejnowskiT.J.,J.Neurosci.,26(2006) p6