it exists in many regions of the brain , such as ca1 of the hippocampus , visual cortex    and cortical areas correlating with conscious perception . it is known that synchronization is very important for information processing , such as predicting sensory input , and information codes . moreover , synchronous activity plays a crucial role in epileptic activity [ 6 , 7 ] , modulation of neurons about attention , memory and learning [ 9 , 10 ] , and cognitive functions . since the discovery of long term potentiation ( ltp ) and ltd ( long term depression ) [ 1214 ] , it has been debated how synaptic modifications are correlated to neuron activities . spike timing - dependent plasticity ( stdp ) is a form of synaptic modification discovered relatively recently , which depends on the relative timing of pre- and post - synaptic action potentials at a millisecond time scale [ 15 , 16 ] . many experiments have proved the existence of stdp , such as in neocortical slices , hippocampus slice , hippocampal cell cultures , and tadpole rectum in vivo . in addition , stdp provides powerful mechanisms   for models of temporal pattern recognition , temporal sequence learning [ 22 , 23 ] , a continuous - time associative memory , coincidence detection [ 16 , 25 ] , navigation [ 26 , 27 ] and direction selectivity . the interaction among neurons relies much on synaptic modification in which stdp is the only one that greatly expands the capability of hebbian learning to address temporally sensitive computational tasks . for example , the result of learning - induced synchronization of a neural network at various developing stages using stdp rule is consistent with recent experimental observations . furthermore , the comparison of synchronization between discontinuous anti - stdp(dc - astdp , see section 2 ) and constant connection has been investigated . following it , the continuous stdp(c - stdp , see section 2 ) has also been studied   by the same authors . motivated by their work , we systemically discuss the roles of four types of stdp rules(c - stdp , dc - stdp , dc - astdp and in - stdp , see section 2 )   in frequency synchronization in the present paper , employing the same model   with only values of some parameters different , such as aplus , tsyn , vslope , gmax ( see section 3 ) . we then consider if the learning curves , which characterize the stdp rule , have certain effects on synchronization . results indicate that synchronization strongly depends on the specific shape and the parameters of the stdp rule . however , the optimal synchronization ranges for dc - stdp and in - stdp , got from regulating learning parameters , are not wider than those for the corresponding strongest constant connection respectively . as a result , when we seek the reason , we discover that the synchronization mechanisms of above four stdp rules can be classified into two categories : ( i ) c - stdp and dc - astdp rules ; ( ii ) dc - stdp and in - stdp rules . the synchronization mechanisms of the two categories are different . for c - stdp and dc - astdp rules , two neurons ' synchronization either relies on the balancing out potentiation and depression during one cycle consistent with the perspective of nowotny et al . or relies on   the maximal synaptic conductance . however , for dc - stdp and in - stdp rules , the synchronization windows are completely provided by the respective maximal synaptic conductance . as regards this finding we consider two hh neurons with unidirectional activity - dependent excitatory or inhibitory synaptic coupling . although such a configuration is too simple to find applications in brain information processing , it serves as a staring point for many model researches . currents , 
 ( 1)cdvi(t)dt=gnami(t)3hi(t)(vi(t)ena)gkni(t)4  (vi(t)ek)gl(vi(t)el)isyn(t)+istim , 
					 where i = 1 , 2 . each of the activation and inactivation variables yi(t ) = { ni(t ) , mi(t ) , hi(t ) } , i = 1 , 2 satisfies first - order kinetics , 
 ( 2)dyi(t)dt=y[vi(t)][1yi(t)]y[vi(t)]yi(t ) ,  i=1,2 . istim is a constant input current forcing each neuron to spike with a constant , istim - dependent period , labeled as t1 and t2 . the postsynaptic neuron would show another firing period t2 , when it is driven by the synaptic current , which is dependent on the postsynaptic potential v2(t ) , the reversal potential vrev , the activation variable s(t ) and its   maximal conductance g(t ) , 
 ( 4)isyn(t)=g(t)s(t)(v2(t)vrev ) , 
					 where 
 ( 5)ds(t)dt = s(v1(t))s(t)tsyn1(s(v1(t))),s(v)={tanh  ( vvthvslope),for  v > vth,0,otherwise . the time - dependent synaptic coupling strength g(t ) ns is 
 ( 6)g(t)=gmax  2(tanh  ( grawgmidgslope)+1 ) . 
  therefore g(t ) always have values between 0 ns and gmax . the bound imposed on g(t ) is artificially set to avoid unrealistically high synaptic conductance and negative conductance . in order to obtain biologically plausible synaptic conductance , several methods have been employed to limit the synaptic strength in literature , such as a negative total integral , artificial bounds , and self - limitation . unless otherwise stated , we employ the self - limitation method which is characterized by a function  tanh  in our simulation . the parameters of the model are 
 ( 7)c=30  f ,  gl=1  s ,  el=64  mv , gna=360  s ,  ena=50  mv ,  gk=70  s , ek=95  mv ,  vth=20  mv ,  tsyn=25  ms , vslope=15  mv ,  gmax  = 25  ns ,  gmid=12gmax  , gslope = gmid ,  vrev=20  mv . the time - dependent synaptic coupling strength g(t ) is determined by the spike - timing of pre- and postsynaptic spikes . we consider four types of activity - dependent couplings that have been found in experiments : ( 1 ) an excitatory synapse with continuous stdp ( c - stdp ) . one ( figure 1(a ) ) is from the recording of the neocortex - layer 5 xenopus tectum hippocampus [ 31 , 35 ] , and the other is from the neocortex - layer 4 spiny stellates . the latter form will not be considered here , because it introduces persistent decrease to synaptic strength that would result in none synchronization if two neurons have different inherent periods . ( 2 ) an excitatory synapse with discontinuous stdp ( dc - stdp , figure 1(b ) ) [ 17 , 36 ] ; ( 3 ) an excitatory synapse with discontinuous anti - stdp [ 36 , 37 ] ( dc - astdp , figure 1(c ) ) ; ( 4 ) an inhibitory synapse with stdp ( in - stdp , figure 1(d ) ) [ 18 , 19 , 36 ] . graw is a function of t = tpostspike  tprespike , time difference between the times of postsynaptic and presynaptic spikes . the learning rules corresponding to figures 1(a ) , 1(b ) , 1(c ) , 1(d ) ) are provided as follows 
						 ( 8)c - stdp:graw={aplust0tpluset / tplus,t>0,asubt0tsubet / tsub,t0,dc - stdp:graw={apluset / tplus,t>0,asubet / tsub,t0,dc - astdp:graw={apluset / tplus,t>0,asubet / tsub,t0,in - stdp:graw={aplus(et / tplus0.5),t>0,asub(et / tsub0.5),t0 ,  aplus = asub . synchronization of pre- and post - synaptic neurons occurs when |t1  t2| is limited in an acceptable range . although there is some arbitrariness in setting the criteria of synchronization , there is no qualitative change in our results if the criteria change in two folds . we have observed that simulations from different initial values of v2 , s could result in different outputs , that is , the post - synaptic neuron sometimes synchronizes with the pre - synaptic one , sometimes keep its original period , or sometimes fires with an oscillating period ( see figure 2(c ) ) . we therefore carry out 40 times of simulations , from randomly selected initial values , for every t2 . the standard deviation of |t1  t2| , indicating how precisely the neurons are synchronized , represents the quality of synchronization . range of t2 , in which post - synaptic neuron is successfully entrained by the pre - synaptic neuron , that is , |t1 t2| < 1.5 in all 40 simulations , is defined as the synchronization window . we investigate the width of synchronization window of various stdp curves , with the same set of parameters . the period of the pre - synaptic neuron is chosen to be 171 milliseconds , which falls into the range of theta waves . first , it has fairly wide synchronization windows which allow comparisons in a relative precise manner and can provide clearer information about synchronization windows of various stdp learning rules . second , the slow theta waves always involve many neurons that fire synchronously [ 38 , 39 ] . also , theta waves have many interesting implications . for example , theta waves are normally absent in healthy awake adults , but appear during the state of meditation .   during emotional arousal and various types of rhythmic activities during sleep , and it is known that coherent theta activity ( 48 hz ) in amygdala - hippocampal circuits is deeply involved in fear memory . with fixed period of the pre - synaptic neuron t1 , we evaluate the coupled period of postsynaptic neuron t2 when it is driven by the pre - synaptic neuron . the values of learning parameters used in c - stdp , dc - stdp , dc - astdp are aplus = 9 ns , asub = 6 ns , tplus = 100 milliseconds , tsub = 200 milliseconds , additionally 0 = 30 milliseconds in c - stdp , and in in - stdp are   aplus = asub = 8 ns , tplus = 100 milliseconds , tsub = 200 milliseconds . our model and most values of parameters are from the model of nowotny for c - stdp , except aplus , tsyn , vslope , gmax are different . especially , t1 is fixed at 171 milliseconds in our simulations while t2 is set to constant value 300 milliseconds of nowotny 's work . the window of synchronization ( upper panel ) and quality ( middle panel ) of dc - stdp are presented in figure 2 as an example . the upper panel shows the number of synchronization times in 40 simulations , for each t2 . it is clear that , in certain range of t2 , simulations from different initial values may have different results . only when t2 falls into the segment from 194 to 221 , the post - synaptic neuron can synchronize with the pre - synaptic neuron from any initial value . it is easily found that there are some t2 corresponding to the number of synchronization times between 1 and 39 . in this situation , we present the three possible states of post - synaptic neuron 's firing in figure 2 ( lower panel)keeping the initial period ( squares ) , oscillating ( circles ) , and synchronizing with the pre - synaptic neuron ( dots ) . obviously , these states are independent of the synchronization criteria we set . for the purpose of discussing the function role of stdp rules in synchronization , the synchronization windows of various type of stdp are plotted in figure 3 , in which the case of constant synaptic conductance is also included as a comparative tool . the same parameter values used in simulations ensure a fair comparison . in our simulation studies , we choose the maximal synaptic strength and the middle synaptic strength of stdp synapse as the synaptic strength of constant synapse in this study . interestingly , because the synchronization windows for stdp rules are narrower than synchronization window for constant synapse g = 25 ns in figure 3 , these results , opposite to previous reports , indicate that all stdp rules do not enhance synchronization comparing with the constant synapse under the chosen parameters . first , increasing the excitatory constant synaptic connections from 12.5 ns to 25 ns , leads to a wider synchronization window . second , the widest ranges of synchronization window are achieved by the excitatory constant connection g = 25 ns and c - stdp rules . however , the lower boundary of the synchronization window of c - stdp is much nearer to t1 than that of excitatory constant synapse . thirdly , although c - stdp and dc - stdp are fitted from the same set of experimental data , c - stdp has a much wider synchronization window than dc - stdp . we conclude that all stdps do not give rise to enhanced synchronization and the window of dc - stdp is surprisingly narrow under the chosen parameters in figure 3 . therefore , the questions about what bring about these results inspire us to study further . at the same time , figure 2 shows a large part of probabilistic synchronization , whose range is a subset of the synchronization window of constant synaptic connection with g = 25 ns . we are interested in if the probabilistic synchronization could be enhanced into absolute synchronization by modulation of learning curves . these are the theme of the next section . in order to establish the functional role of stdp clearly , we consider if the learning parameters for each stdp rule have important effect on synchronization . an increase in the degree of synchrony of a uniform input can cause transitions between memorized activity patterns in the order presented during learning . however , if synchronous input is at a low level , transitions can not occur . the synchrony - asynchrony transition have also been implemented in controlling winner - take - all competition , the next recalled time of associative memory   and the fine structure of cell assemblies . in this section , we will discuss the flexibility of the synchronization window , by exploring regulation of width of the synchronization window , whose boundary indicates the synchrony - asynchrony transition . we take the modulation of learning parameters as the method to regulate the synchronization window . there are four parameters that determine a learning rule aplus , asub , tplus , tsub . with three of them fixed and only one parameter changing , we could explore its influence on the width of the synchronization window . for example , aplus increases from an adequately small value 1 ns to 20 ns with a step of 1 ns . we have also scanned values that are beyond 20 ns , but find that the effect of increasing aplus is saturated around 20 ns . further increasing aplus brings no more effect . other parameters are fixed as : asub = 6 ns , tplus = 100 milliseconds , tsub = 200 milliseconds , t2 = 233 milliseconds . in these conditions , we present the effect of aplus on the location of the synchronization window with the dc - stdp rule . figure 4(a ) shows the value of arp ( average change of relative period ratio ) = t2  t2/(t2  t1 ) for different aplus . some points have value 0 or 1 , which means post - synaptic neuron keeping initial period or achieving synchronization with the pre - synaptic one , respectively . figure 4(b ) gives an explanation that these points correspond to probabilistic synchronizations with fixed t2 = 233 . we find that the absolute synchronization range is from 10 ns to 20 ns in figure 4(b ) . according to the definition of synchronization window of t2 , we can similarly define 10 ns to 20 ns as the synchronization window of aplus , with dc - stdp rule and other fixed parameters . the boundary of this synchronization window indicates where synchrony - asynchrony transition happens when changing aplus . from the results of figure 4 , the reason why synchronization window of constant synapse is wider than that of stdp rules ( figure 3 ) may be explained by learning parameters . to figure out a global picture of the effect of aplus on synchronization for c - stdp rule aplus increases from 1 ns to 20 ns with a step of 1 ns while other three parameters keep initial values : asub = 6 ns , tplus = 100 milliseconds , tsub = 200 milliseconds . the points in figure 5(a ) show the synchronization range of aplus with t1 divided by the chosen values of t2 . the lower boundaries , as well as those upper boundaries that are other than 20 ns , indicate the position of synchrony - asynchrony transitions . for example , when t2 is 177 milliseconds equivalent to t1/t2 = 0.966 , aplus outside of the points range from 8 ns to 11 ns can not lead to synchronization between the two neurons . in addition , for those points marked on the horizontal axis , synchronization could not be established no matter what values aplus take . according to this global picture , the intersection of those synchronization ranges of aplus , which is from 10 ns to 11 ns , identifies the range of aplus that would lead to the widest synchronization window which is from 177 to 289 . this optimal synchronization window for c - stdp is wider than constant synapse g = 25 ns . taking the same method as aplus , we study the effects of other three parameters on synchronization . figure 5 presents the situation of asub varying from 1 ns to 20 ns with the step 1 ns ( circles , figure 5(a ) ) , tplus ( points , figure 5(b ) ) and tsub ( circles , figure 5(b ) ) both varying from 10 milliseconds to 400 milliseconds with the step 10 milliseconds . when changing one parameter to explore how the range of synchrony evolves with t2 , other three parameters keep their initial values as in figure 3 . similarly , the global modulation picture of four learning parameters for other learning rules can be got . the optimal synchronization windows for various learning rules are presented in figure 6 comparing with previous synchronization windows obtained in figure 3 . the parameters are derived according to regulating one parameter while other three parameters keep initial values . under the optimal parameters , the synchronization windows for stdp rules are not narrower than constant synapse g = 25 ns . consequently , the reported important role of stdp in synchronization should be dependent on learning parameters . for the first three stdp rules , the optimal synchronization windows are got by regulating tsub while other three fixed learning parameters keep initial values . for the last stdp rule , , we find that the optimal synchronization windows for c - stdp rule and dc - astdp rule are almost equal , and are wider than constant synapses . however , comparing with these two rules , the optimal synchronization window for dc - stdp rule is much narrower . for inhibitory stdp , the synchronization window is the same as that for constant synapse under connection strength g = 25 ns . accordingly , what makes the optimal synchronization windows for various stdp rules different deserves an explanation . we find that the mechanisms of synchronization caused by c - stdp ( or dc - astdp ) and dc - stdp ( or in - stdp ) are different in our model . figures 7(a ) and 7(b )   show the average t = tpostspike  tprespike , and synaptic strength after an episode of coupling time for c - stdp rule . there are two types of behavior for t when synchronization occurs ( figure 7(a ) ) . in a section of constant t , the synaptic strength does not achieve the maximal value . apparently , in this situation , postsynaptic neuron achieves synchronization with the pre - synaptic neuron depending on the balance between potentiation and depression of synaptic conductance . in the rest part of synchronization window it indicates that for larger t2 , postsynaptic neuron achieves synchronization depending on the effect of maximal synaptic conductance . for dc - astdp rule , when the post - synaptic neuron synchronizes with the pre - synaptic neuron under small t2 , the change of synaptic potentiation and depression cancel each other . however , for the small portion of synchronization window at the right side , synaptic conductance gets the maximum at the stationary synchronized state . for dc - stdp and in - stdp rules , the synchronization mechanisms may be different with the above two stdp rules . because the potentiation and depression of synaptic conductance cancel each other however , for the dc - stdp rule ( figures 7(c ) and 7(d ) ) , t keeps varying which means that the potentiation and depression of synaptic conductance do not cancel each other at the synchronization state . but , it is easily found that the synaptic conductance is at the stationary maximum for dc - stdp rule . thus , postsynaptic neuron achieves synchronization completely depending on the effect of the maximal synaptic conductance for dc - stdp rule . the state of in - stdp rule is similar to dc - stdp rule . as a result for the few t2 at the right side of the synchronization window , the synaptic conductance achieves the maximum with c - stdp and dc - astdp rules . because the frequency mismatch is larger , the synapse needs to be stronger to entrain the post - synaptic neuron . but for most part of synchronization window at the left , c - stdp and dc - astdp rules rely on the balance of potentiation and depression . instead of balancing out potentiation and depression during one cycle , dc - stdp and in - stdp rules depend on synaptic strength achieving its maximum . it is important to understand why c - stdp and dc - astdp rely on the balance of potentiation and depression while dc - stdp and in - stdp do not . have introduced the mechanisms behind the enhancement of neural synchronization by c - stdp rule which rely on the balance of potentiation and depression . the synapse strength remains stable regardless of postsynaptic neuron firing later or earlier attributed to the specific shape of c - stdp curve . we adopt the similar analysis method   here for the dc - astdp and dc - stdp . the time lags are recorded as t1 and t2 , where t1  t2 = t1 = t2 and g1  g2 = 0 ( figure 8) at this state . synaptic strength will be depressed , due to g1  g2 < 0 for dc - astdp rule , so that the post - synaptic neuron is less excited and goes back into the synchronized state ( figure 8(b ) ) . but for dc - stdp , when post - synaptic neuron fires faster , synaptic strength will be increased due to g1  g2 > 0 in this case . the post - synaptic neuron is more excited and can not go back into the synchronized state ( figure 8(c ) ) . the opposite direction is the same case for dc - stdp and can not go back into the synchronized state . therefore , the synchronization mechanisms between these two rules are different . for in - stdp rule , we can easily find that g1  g2 is always positive , where g1  g2 = aplus  ( exp(x / tplus)0.5 )  asub  ( exp(x171/tsub ) 0.5 ) and values of parameters are the same as figure 3 . it means that the potentiation and depression of synaptic conductance during one period can not achieve balance . the synaptic strength must achieve the maximum resulting from g1  g2 > 0 . on the other hand we try to explain the role of a learning parameter by considering how it influences the synapse conductance , which is a major factor for synchronizing neurons with a given mismatch of intrinsic frequencies . it is obvious that if synaptic conductance becomes stronger , it can make larger t2   to achieve the same period with pre - synaptic neuron for various stdp rules . thus , with other three learning parameters fixed , larger aplus values , which are corresponding to the stronger stable synaptic strength for c - stdp rule , can cause larger t2 synchronization . similarly , smaller asub values will give rise to larger t2 synchronization for c - stdp ( figure 5(a ) ) moreover , from figure 5(b ) , moderate tplus values can also make larger t2 synchronized to t1 because these values bring about stronger synaptic strength . based on the expression of c - stdp rule in section 2 , let tplus be variable and let other parameters keep constant . graw is viewed as the function of variable tplus . by calculating the derivative of graw , we can find that graw is a first increasing and then decreasing function when tplus increases gradually . thus , medial values of tplus can result in stronger synaptic strength . by the same reasoning for tsub , we can conclude that smaller or larger tsub can make larger t2 synchronization for c - stdp ( figure 5(b ) ) . the effects of learning parameters on synchronization about other learning rules are similar to the c - stdp rule for larger t2 . however , smaller t2 values leading to synchrony are only related to c - stdp and dc - astdp , because they rely on the balance of depression and potentiation which could lead to an appropriate low stable synaptic strength . therefore , only proper learning parameters got by regulating the effect of learning parameters on synchronization are required for smaller t2 achieved synchronization . finally , we conclude why the widest synchronization windows for some stdp rules are different . from the above statements the other is the balance of depression and potentiation which can make smaller and moderate t2 synchronize . for c - stdp and dc - astdp however , dc- stdp and in- stdp rules , due to their specific shape , can only make use of the first one . this implies that the widest synchronization windows for dc - stdp and in - stdp rules can not exceed the synchronization windows for the maximum constant connection strength under the same model respectively . therefore , the optimal synchronization windows for c - stdp and dc - astdp are wider than those for dc - stdp and in - stdp . inspired by previous experiments and theoretical researches , we study the important aspects of stdp - induced synchronization in this paper , such as the role of various stdp rule in synchronization , the widest synchronization window through regulating learning parameters , and synchronization mechanism . in order to explore the functional role of stdp in synchronization , we compare synchronization windows of different types of stdp rules with that of constant synapse under the same model parameters . for the given parameters , synchronized responses have a stronger influence on cells at subsequent processing stages than nonsynchronized responses [ 4850 ] . and the enhanced precise synchronization is important in improving a rapid and reliable transmission of information about sensory changes [ 2 , 51 ] . recent researches have reported various methods to enhance synchronization , such as , selective attention   and time delay . here , we present the effect of modulation of learning parameters on synchronization and optimal synchronization window which are not narrower than constant synapse . the optimal synchronization windows by c - stdp and dc - astdp rules are much wider than constant synapse . it indicates that the function role of stdp rule in synchronization depends on the learning parameters . the optimal synchronization windows of c - stdp and dc - astdp are wider than that of dc - stdp for excitatory synapse . for c - stdp rule and dc - astdp rule , a stationary synchronized state completely depends on the balance between potentiation and depression or the maximal synaptic conductance . however , for dc - stdp and in - stdp rule , the stable synchronized state depends on the maximal synaptic conductance under the self - limitation of synaptic strength . if we change the type of bound of synaptic strength for dc - stdp rule from self - limitation to artificial bounds , we find that the synchronization mechanism does not change . in a word , on one hand , the synchronization range of dc - astdp and c - stdp can achieve the optimal synchronization window of dc - stdp , depending on the maximal synaptic conductance . on the other hand , dc - astdp and c - stdp can extend the synchronization windows to include smaller t2 by the balance between potentiation and depression . for example , neurons in cat area 17 can be grouped in 4 different electrophysiological cell classes , including regular spiking . and spontaneous , regular action potentials were observed both with cell - attached patch recordings as well as with whole cell current - clamp recordings for cholinergic neurons in the parabigeminal nucleus of the rat midbrain . for example , whether pyramidal neurons in different cortical layers exhibit similar tendencies to synchronize is studied . based on this point and the functional role of stdp in synchronization , we explore the synchronization windows of various stdp rules from the view of neurons ' regular firing . stdp - mediated synchronization is a remarkably robust phenomenon against strong noise [ 30 , 31 ] . although our simulation is not under the noise environment , our results may represent some predictions for stdp - mediated synchronization in noisy environment . in addition , from figure 2(a ) , we can clearly see that some synchronization number is between 1 and 39 . our results present that the range of t2 values leading to synchrony increases strongly if the constant synaptic connection is increased from 12.5 ns to 25 ns . find that the extent of synchrony does not change considerably by doubling the synaptic conductance . furthermore , t1 is fixed at 171 milliseconds in our simulation while t2 is fixed at 300 milliseconds in theirs . when t2 is fixed , the range of t1 values leading to synchrony is limited from 0 to t2 no matter how the strength of constant connection changes . however , when t1 is fixed , the range of t2 values leading to synchrony can vary from t1 to very large value due to the increase of the constant connection . at the same time , if we adopt the same parameters with nowotny 's paper , the similar result can be got . furthermore , when constant synaptic connection is 0 ns , it is clear that two neurons with different initial periods can not synchronize . along with the increase of strength of constant connection , some t2 must cause the synchronization . thus , it is easily found that synchronization window must become wider by increasing the constant connection to some degree . the question of how the time windows of various stdp rules are biophysically regulated remains relatively unexplored . it has been proposed that conscious perception depends on the transient synchronization of widely distributed neural assemblies . and long - distance synchronization plays a role in triggering the cognitive processes associated with conscious awareness . in addition , some diseases and the function of brain are related with synchronization mentioned above , especially in theta ( 48 hz ) rhythm synchronization during fear memory retrieval   which is consistent with what we considered here .