oscillation is ubiquitous in nature , not only in physics and chemistry but also biology .biological oscillations can be observed over a wide range of time- and population - scales , from a circadian rhythm of about 24 hours  to a segmentation clock of less than 2 hours  , from whole - body oscillatory fevers  to periodic protein production in a single cell  . on the other hand ,sound theoretical studies have been undergoing since long before the observation became possible in molecular level .there are many theoretical models to explain these phenomena . despite their diversity of biological insights , these models share some common points .proteins are produced by the transcription and translation of specific sequences of dna .on the other hand , proteins can bind to a transcription promotor on dna and hence suppress or enhance gene expression .a transcriptional negative feedback loop  and a delay  in the inner cellular gene - protein network are considered to be important elements that contribute to the oscillatory expression of dna and protein production . from the perspectives of dynamical systems , such oscillations are limit cycles that can be generated from hopf bifurcation by choosing an appropriate parameter set and initial condition .consequently , in the case of a cell group or multi - cellular organism with an oscillatory character , such as cardiac tissue and a segmentation clock in the tail of psm ( presomitic mesoderm ) , the synchronization of coupled oscillators is often used to explain the observed collective oscillation  .however , periodic oscillation is only a small part of the dynamical behavior of a cell .oscillation may cease if the conditions are changed , and most cells tend to settle into a seeming stable state .for example , electrical activity in -cells exhibits slow periodic oscillation at the macro scale of islets of langerhans , while much faster excitability instead of oscillation when isolated  . in another example, the three proteins ( kaia , kaib , and kaic ) , identified as important for the circadian rhythms in cyanobacterium _ synechococcus elongates _ , behave as a bistable toggle switch due to a double - negative - feedback loop .oscillation could then arise from the successive switch between these two stable steady - states  .moreover , most recent studies also suggested that negative transcriptional feedback is not sufficient , and in some cases not even necessary , for circadian oscillation .instead , intracellular signaling , such as that involving ca and camp , together with transcriptional feedback plays a key role in long - term circadian pacemaking  .these evidences raise the possibility that intrinsic oscillatory cells are not indispensable in an oscillatory organism .  in this paper , we study the occurrence of collective oscillation from non - oscillatory system .in contrast to the conventional mechanism of synchronized oscillators , none of the individual cells in our model is intrinsically oscillatory .a few studies in the context of mathematics and physics have revealed the possibility of collective oscillating patterns .the first example was proposed by smale  , who found that two `` dead '' cells can become `` alive '' via diffusive coupling .more recently , other studies have examined this behavior in detail  . in - phase and anti - phase self - sustained oscillation of excitable membrane via bulk couplinghave been observed  .the models considered in these reports have mostly involved coupled identical excitable cells with mono - stability .some more complicated approaches include , for example , using a unidirectional coupling scheme  , applying a periodic stimulation  , coupling the system with an oscillatory boundary  , introducing heterogeneity into excitable media  , activity propagating in discrete cellular automata model  , and so forth .a commonly used idea is to set isolated cells at a subthreshold quiescent state , and then push them over into the oscillatory regime to generate pacemaker cells by extra force or coupling .that means cells are possible to manifest themselves as oscillators .however , little attention has been paid to the emergence of oscillation in systems that are completely independent of oscillating elements . unlike previous studies , geometrical structure of nullclines of cells in our model prevent dynamics from being oscillation .there are two `` engines '' in our model to drive the self - sustained collective oscillation , neither of which is oscillatory pacemaker .the one is bistable cell switching between two stable states , the other is mono - stable cell with excitability .two engines work cooperatively due to the wave propagation .  on the other hand , in a bistable system, a stationary front can bifurcate into a pair of fronts that propagate in opposite directions , which is known as non - equilibrium ising - bloch ( nib ) bifurcation  .perturbation for the occurrence of nib bifurcation can be induced by local spatial inhomogeneity  .a more global analysis showed that the nib point is only part of the story , and concluded that an unstable wave front is intrinsic to media that are spatially inhomogeneous  .an unstable wave front may manifest itself as a reflected front , tango wave  , pacemaker  and so on . in this paper , we think about these phenomena beyond mathematics and physics , and extend their application to biological oscillators .  moreover ,although most studies have been performed on a spatial continuum described by partial differential equations ( pde ) , continuum models neglect the effects of cellular discreteness  .in fact , from the viewpoint of biology , the size of cells can not decreased infinitely .this intrinsic property is difficult to ignore , especially at the stage of initial development of an organism , when the cell size is comparable to that of tissue .in addition , there are mathematical reasons to explore the system dynamics with spatial discretization .pde and ode ( ordinary differential equations ) have different theoretical frameworks and produce different results .several significant features of discreteness , such as _ wave propagation failure _ , can not occur in a continuum model .therefore , in this paper we will consider an array of spatially discrete cells , and discuss the impact of discreteness .in this paper , we consider cells in one - dimensional space .cells are coupled by intracellular signaling molecules , which flow through channels in a membrane due to concentration difference or depolarization - mediated flux .the intracellular signaling small - molecule can be produced by a series of process from some genes functioned as activator , and then trigger transcriptional feedback loops of adjacent cells .we assume that the coupling interaction takes place in a diffusion - like manner .if we include an inhibitor , which can locally repress the expression of activator genes , a one - dimensional array of cells can be described as where and are concentration of activator and inhibitor , respectively , is the index of the cell in the chain and is the coupling strength of . and are corresponding reaction functions .the boundary condition is zero flux , i.e. , and .finally , is an environmental parameter , which will be discussed in detail later .  in this study , we only consider coupling of the activator . for most of the models that have been used to study pattern formation , diffusion is assumed to occur for every elements .specially , much greater diffusion of the inhibitor is necessary to induce turing instability  .however , a cells membrane is very selective for passage of substances .complicated intracellular reactions usually take place locally , but are triggered by only one or a few specific signaling molecules . for example , while the segmentation clock involves the cyclic expression of many genes , the crucial pathway for coupling only involves the transmembrane receptor notch1  . thus ,in the context of biology , we only consider coupling with the activator , and the inhibitor in our model is merely a local state variable .the development of a multicellular organism begins with a single cell , which divides and gives rise to cells with different typologies .different cells are organized according to certain secreted chemicals , called morphogens . despite improvements in experimental and theoretical approaches , the mechanisms of morphogenesis are still unclear .usually , morphogens are considered to be produced at specific sites and diffuse through the organism  .quite recently , evidence of a `` shuttling - based '' mechanism has been presented  .the key in such models is their ability to define a robust and scaling profile , usually a concentration gradient , of morphogens .more broadly , we can suppose that some environmental parameters act as morphogens .the environment in which an organism develops supplies nutrition for growth , and the intracellular volume in direct contact with the border gets more and that deep inside cells gets less .  in this paper, we do not consider any specific chemical substance , and instead merely suppose that there is a certain factor , which we refer to as the _ active factor _ , to obtain information regarding the relation between position and cell dynamics .the above - mentioned active factors can affect the fate of cells in a concentration dependent manner  .  without losing generality , we assume that the active factor is constant at the boundaries of an organism , where the source site of morphogens are usually located .it diffuses into the organism field with a diffusion constant , and is degraded at rate .thus , we have since our model is based on coupled odes independent of spatial variation , the profile of the active factor satisfies a scaling property . by normalizing the field size to one ,we can get a steady profile ( ) of as where is the value at two boundaries , and is the inverse of the decay length . a typical profile of is shown in fig .[ fig : gamma ] .circles indicate the value of for discrete cells ( in the figure ) placed uniformly in the scaling field .obtained from eq.([eq : gamma - profile ] ) , when . ]we assume that cells normally prefer to live in a stable state , and cells with a high concentration of active factor are capable of switching between two states .this kind of bistability is very important and has been observed in various biological systems  .for example , the expression of the _ dictyostelium _ camp phosphodiesterase gene behave as a bistable switch employing intracellular camp as a regulator of cell fate  , the cdc2 activation system in xenopus egg extracts is bistable and characterized by biochemical hysteresis  , the inducible _ lac _ operon in _e. coli _ shows bistability  , and so on .usually , bistability arises from positive , or double - negative genetic regulation loops  .it was recently suggested that stochastic fluctuation plays an important role in the nature of the transition between bistable states  .moreover , physical regulation of protein production , which has been much less considered by biochemists , also plays an important role in the origin of the bistability .it has been observed that discrete transition between folding and unfolding states , namely a first - order phase transition , can take place in giant dna  .similar discrete switch can also occur for rna  , protein  and other molecules  .this discrete transition leads to the on / off switching of the production of a specific protein .we describe the dynamical reaction function of each cell by using the two - component fitzhugh - nagumo equations where is a variable related to the expression level of specific activator genes , is the inhibitor to repress , , which is much smaller than 1 , is the slower growth factor of inhibitor and is the active factor discussed previously . note that the kinetics of inhibitor here is a rather natural unit process in many of biochemical reactions . throughout this paper ,the following parameters are fixed the fitzhugh - nagumo model has been well studied for description of excitable behavior in biology .rich nonlinear dynamics can be observed by tuning parameters .specifically , with the above parameters , the model is mono - stable at small value of , and will happen a saddle - node bifurcation at and a hopf bifurcation at , which leads to bistability .thus , in the case of the spatial profile of as shown in fig . [ fig : gamma ] , only the 8 central cells ( 7 - 14 ) are mono - stable , while the others are bistable ( ) .figures [ fig : onoff](a)&(b ) show the nullclines with bistability and mono - stability for when is large and small , respectively .again , none of the cells show oscillation in the absence of coupling .more important , from the geometry property of nullclines in the figure , no oscillatory condition could be found by moving cubic nullcline ( ) up and down .that is , no pacemaker cells can be generated from activator coupling .if we substitute eq .( [ eq : fhn1]&[eq : fhn2 ] ) into eq .( [ eq : chain1]&[eq : chain2 ] ) , we get the system equations used in this paper . when we change the coupling strength , we observe the occurrence , variation and disappearance of self - sustained collective oscillation in the cell array .in eqs . ( [ eq : sys1 ] & [ eq : sys2 ] ) .( a ) spatio - temporal plot of the collective oscillation of .the black and white indicate and , respectively .( b , c ) waveforms of ( solid ) and ( dashed ) in the 3rd and 4th cells . ]  figure [ fig : normal_osc](a ) shows a typical oscillation when the coupling strength .figure [ fig : wave-1p](a ) shows an enlarged view of a single period of oscillation . as the initial condition, we set the 1st cell as being excited , since stimulation is usually input from the border .initially , ( ) , a traveling wave appears due to excitation at the border .the traveling front then sweeps over the cell array and makes all of the cells excited ( see fig .[ fig : wave-1p](b ) ) .although the central cells are also turned on due to the interaction with other cells , they can not stay in the excitable state for a long time .instead , they soon return to their stable equilibrium ( see fig . [fig : wave-1p](c ) ) , and hence generate two counterpropagating wave backs , as shown in fig .[ fig : wave-1p](d ) .these two wave backs propagate outward until the 3 and 18 cells and stop suddenly due to the spatial discreteness ( see fig .[ fig : wave-1p](e ) ) .the `` wall '' cells do not jump from the on state to the off state and only exhibit slight oscillation closed to their equilibrium . as an example, the difference between the 3 and 4 cells is shown in fig .[ fig : normal_osc](b , c ) . at this critical interface ,the inhibitor slowly decreases so that the 4 and 17 cells restore excitability after a while .the central cells can then be excited again by the pair of reflecting wave fronts , as shown in fig .[ fig : wave-1p](f ) . pushed by the wave , the central cells will be excited again .this process repeats and causes the collective oscillation inside multi - cell tissue without oscillatory cells .   over a single period ,( b)-(f ) snapshots of and at several time points in one period , where the horizontal axis is cell number from 1 to 20 .this illustrates the change in wave propagation at different stages .solid curves and dashed curves indicate and , respectively .an animation , through which the behavior can be understood more intuitively , is available at . ]    in a stationary state .wave propagation stops and no oscillation occurs in the case of weak coupling . ]  .( a ) spatio - temporal diagram of .the grayscale black and white indicate and , respectively .( b ) and ( c ) are waveform diagrams of the 3rd and 18th cells . activator and inhibitor are shown in solid and dashed curves , respectively . ]  .( a ) spatio - temporal diagram of .the grayscale black and white indicate and , respectively .( b ) and ( c ) are waveform diagrams of the 3 and 18 cells .activator and inhibitor are shown in solid and dashed curves , respectively . ]the above collective oscillation can be observed when the coupling strength is larger than a threshold , below which wave backs ( see fig . [fig : wave-1p](d ) ) fail to reflect , and the state in fig .[ fig : wave-1p](e ) is maintained .figure [ fig : beforeosc ] shows a spatio - temporal diagram , where the central cells stay silent while excited bands appear close to the two borders .  note that this phenomenon could not take place in a continuum counterpart .the existence of a coupling strength threshold under which _ wave propagation failure _ occurs is unique to a spatially discrete system .in addition , there is another threshold , which is even smaller , for which the _ wave front _ stops propagating . in this case, the excited signal at the border fails to propagate forward , but we would like to postpone this interesting phenomenon on another paper , since it is less related to present work .   with an increase in the coupling strength , the characteristics of oscillation can be changed .figure [ fig : antiphase_osc ] shows that the position of oscillation periodically shifts .the 3 and 18 cells oscillate with a nearly doubled period , in anti - phase ( fig .[ fig : antiphase_osc](b , c ) ) . globally ,tissue oscillates in two groups with the same cell populations but different positions : no . 3-no .17 ( 15 cells ) and no .4-no . 18 ( 15 cells ) , respectively .interestingly , by slightly increasing the coupling strength , say to , we found a different type of period doubling , as shown in fig .[ fig : inphase_osc ] . for comparison with the case of , although the critical interface between on and off shifts periodically as in fig .[ fig : antiphase_osc ] , there is no phase difference between the 3 and 18 cells . as is clearly shown in their waveform ( fig .[ fig : inphase_osc](b , c ) ) , these two boundary cells oscillate in - phase , instead of anti - phase ( fig .[ fig : antiphase_osc](b , c ) ) . therefore , in the present condition , a periodic change does not take place in the position of oscillation .instead , the population of oscillating cells changes .more precisely , tissue oscillates in two groups : no .18 ( 16 cells ) and no . 4-no .17 ( 14 cells ) , respectively .  moreover , by setting the initial condition of the cells identically , i.e. , all in the on state at , we found checked that same symmetric collective oscillation can also occur in the case of . therefore , we conclude that these two types of oscillation are caused by the same bifurcation .because the wavefront propagates faster with larger , a larger coupling strength can reduce the time lag between the two boundary cells being stimulated .if the time lag is smaller , the two boundaries converge to in - phase oscillation . on the other hand , if the time lag is large , they will exhibit anti - phase oscillation .   .black and white indicate and , respectively .( a ) , oscillation starts to collapse ; ( b ) , oscillation ceases after one cycle . ]  with a increase in coupling strength , we observed that the change in the periodic position or population stopped , and normal oscillation returned . in comparison to the case of ,the total population of oscillating cells increased from 14 ( no . 4 to no .17 ) to 16 ( no . 3the oscillation suddenly dies when is as large as 2.6 .figure [ fig : osc_death](b ) clearly shows that the central cells start to oscillate after all of the cells are excited , but this oscillation is not sustained . in this strong coupling condition , the boundary cells can not recover their excitability , so that the wave front propagating from the center is unable to stop and reflect to generate successive oscillation .before the oscillation stops , there is a narrow parameter region of , where only one side of the `` wall '' alternatively collapses , and a complicated period-4 collective oscillation is observed ( fig .[ fig : osc_death](a ) ) .there are many factors that may influence the oscillation behavior .for example , if the spatial profile of active factor becomes more `` steep '' rather than a gentle slope , the wave back will tend to be locked and fail to reflect from boundary .moreover , if we reduce the excitability of cells by increasing , it will be more difficult for wave front to propagate cross the center , and only the half part with stimulation can oscillate . since the global oscillation is induced by mutual coupling , we are going to study the oscillation behavior with respect to the coupling strength . here , we sweep from 0.45 to 2.7 , and summarize the variation in the oscillation period and position of the left border of oscillation region .if the coupling strength is smaller than 0.48 , there is no oscillation , and 4 cells from the tissue border are excited while cells 516 are silent .oscillation takes place when the wave back passes the 4 cell at .the border then shifts between 3 and 4 , while in - phase and anti - phase period doubled oscillation occur , roughly between .finally , the oscillation reaches a maximum region : from cells 3 to 18 , until is too large for oscillation to occur .figure [ fig : phasediagram](a ) shows the expansion of oscillatory region .variations in the period of oscillation are shown in fig .[ fig : phasediagram](b ) .once the central cells start to collectively oscillate , the period rapidly decreases when the coupling strength increases .the rate of the period decrease gradually slows .the period changes little in the region where is large .this phenomenon occurs because the stationary interval ( fig .[ fig : wave-1p](e ) ) greatly contributes to the period of oscillation .the decrease in the stationary interval significantly shortens the period of oscillation when is small .however , when is large enough , the wave backs reflect immediately without stopping , and the period is determined mainly by the velocity of propagation . therefore , the presented oscillation is robust at strong coupling condition , and tunable at weak coupling case .there is a parameter region ( the curve of period is drawn in dashed curve fig .[ fig : phasediagram](b ) ) in which system undergoes period - doubling bifurcation and the period-1 solution lose its stability .meanwhile , period two solutions appear around this region .we show more details in fig .[ fig : phasediagram](c ) . in the figure , we draw two intervals in a period two solution , by measuring the time when positively cross the section : . circles ( ) and crosses ( ) indicate in - phase and anti - phase solutions , respectively .the in - phase period-2 solution is the result of period - doubing bifurcation , while the anti - phase period-2 solution occurs saddle - node bifurcation .the anti - phase period-2 solution has a wider parameter region than in - phase one , and coexistence with fundamental period-1 solution can be observed in both side of .there are quite complicated bifurcation phenomena specially around the occurrence of period - doubling bifurcation .we have even found period-3 solution ( in - phase one around =0.802 , and anti - phase one around , respectively ) .although they are very interesting in the viewpoint of nonlinear dynamical system , we leave them to our future work , because current paper is going to discuss the possibility of global oscillation and its potential applications .the above phenomena are observed in a spatially discrete system , described by ordinary differential equations .as briefly introduced in sec [ sec : intro ] , this discreteness is important in both a mathematical and biological sense . let us discuss this significance in more detail .the diffusion term in a one - dimensional spatially continuous reaction - diffusion model can be formulated as in its difference version .this type of conversion is a common approach to solving pde numerically .the diffusion rate usually does not change much for a specific substance under constant conditions .thus , if we assume that the coupling is mainly due to the diffusion - like effects of substances , the coupling strength changes in square order with respect to variation of , which biologically corresponds to the distance between cells or the cell size . since the profile of the active factor has a scaling property , it is reasonable to suppose that this gradient works for a field of any size .thus , we can study how a change in number of cells and distant of cells affects global dynamics .  .black and white indicate and , respectively .( a ) .( b ) . ]figure [ fig : cell200](a ) shows spatio - temporal diagrams with a 10-fold increase in the number of cells , .other parameters are the same as those in fig .[ fig : normal_osc ] . obviously , more time is required for a wave to sweep over the organism .the period of oscillation and the phase difference between the two sides increase greatly . on the other hand ,if the distance between cells becomes smaller and smaller when cell population increases , the system manifests itself more like a continuum than a discrete system . in this case , the coupling strength will increase dramatically as a square with respect to the decrease in , i.e. , a 100-fold increase in in present case .when is as large as 70 , we have the spatio - temporal diagram given in fig .[ fig : cell200](b ) .when we compare this with fig .[ fig : normal_osc ] , there is little change in the period of collective oscillation .this suggests that the clock tends to run more punctually . in a mathematical sense ,when the population of cells is large enough in a fixed field , the behavior of the organism will follow the solution of a specific partial differential equation , which is independent of the number of cells .[ fig : cell200](a ) simply corresponds to the case that cells grow in an open space , and extend the field by keeping size and distant of cells constantly . during the initial period of development , however , cell divisions within the egg proceed quickly , without much increase of the total cell mass and size .thus , cells at this stage rapidly decrease in diameter .this may interpret biologically the situation of fig .[ fig : cell200](b ) .there are many biological situations , however , that the intercellular coupling does not follow the diffusion - like ways , such as communication involving the delta - notch signaling pathway  . in those cases, has few direct influences on , which may represent `` bottlenecks '' irrespective to the diffusion .thus , the modeling based a continuum is inappropriate for some conditions .   .straight lines are the nullcline of .circles are the position of at specific times .an animation of the dynamical nullclines can be found at . ]self - sustained collective oscillation is caused by the excitability of cells and their mutual interaction .the system involves complicated bifurcations .we present here some qualitative ideas regarding how this oscillation takes place .  from dynamical equations ( [ eq : fhn1],[eq : fhn2 ] ) and their nullcline shown in fig .[ fig : onoff ] , we know that a single cell can exhibit either bistability or mono - stability . however , if we introduce coupling , the geometry of nullclines of one cell will dynamically change according to its own state and those of its neighbors . because it was assumed that the communication between cells is only mediated via the activator ,the strict nullcline is independent of coupling . from eqs.([eq : chain1]&[eq : fhn1 ] ) , we obtain the function for the nullcline as where is the offset of the cubic function due to coupling .thus , the nullcline dynamically moves up and down in the phase plane , corresponding to the state of .  in fig .[ fig : nullclinemove ] , we show the phase portrait of cells around the oscillation border ( cells no.3 - 5 ) , as well as their dynamical nullcline at some turning points .snapshots are taken under the same conditions of normal oscillation as shown in fig .[ fig : normal_osc ] .  the first row , ( a , b , c ) of fig .[ fig : nullclinemove ] are all in the excited state , i.e. , for all three cells , is close to 1 . therefore , under this condition, the vertical offset of the nullcline is nearly zero , and all three cells exhibit bistability .the second row is taken at , when the wave back comes ( see fig .[ fig : wave-1p](d ) ) , and the 5 cell moves towards its lower equilibrium ( fig .[ fig : nullclinemove](f ) ) . since ,a sudden drop in leads to a rapid decrease in the cubic nullcline . as shown in fig .[ fig : nullclinemove](e ) , the cubic nullcline moves down so that the higher equilibrium disappears .thus , the 4 cell becomes mono - stable and the state quickly converges to the left branch of the cubic nullcline .the decrease in pushes back to a positive value , and makes the nullcline of the 3 cell move down , as shown in fig .[ fig : nullclinemove](g , h ) . however , since the 3 cell has a larger , which controls the amplitude of the cubic nullcline , even if decreases to its lowest value ( fig .[ fig : nullclinemove](h ) ) , i.e. , reduces to its minimum , the cubic and straight nullclines still intersect , and the higher equilibrium remain .this explains why the wave back passes the 4 cell , but stops at the 3 cell ( fig .[ fig : wave-1p](a ) ) .after propagation stops , there is a relatively long refractory period from time 230 to 300 . in this interval, there is a slow decrease in the inhibitor . since and , although slight increase in pulls the cubic nullcline down , is still so large that the cubic nullcline is above the straight nullcline ( fig .[ fig : nullclinemove](k ) ) . under this condition ,the cell is mono - stable , with the equilibrium at the right branch of the cubic function .thus , after a while , the state of will switch to a higher value ( fig .[ fig : nullclinemove](n ) ) , and leads to a reflecting wave front ( fig .[ fig : wave-1p](f ) ) . finally , the states return to the situation of fig . [fig : nullclinemove](b ) after another refractory period .note that a smaller coupling strength leads to a smaller offset .if we move down the cubic nullcline slightly to cross the straight nullcline in fig .[ fig : nullclinemove](k ) , the 4 cell becomes bistable .this will disable the switch from left to right , and stop the oscillation ( fig .[ fig : beforeosc ] ) .  from the above description and fig .[ fig : nullclinemove ] , we conclude that the boundary cell , here no.4 , which is bistable without coupling , turns to switch between two types of mono - stable dynamics . as introduced in sec . [sec : intro ] , it is different from the studies changing nullclines via coupling to an oscillatory geometry .this switching becomes the power that underlies the self - sustained oscillation observed in the present model .the variation of the offset of the dynamical nullcline of the boundary cell gives rise to rich oscillation phenomena .we will now explore the conditions for oscillation in an approximate manner by studying the dynamics on the oscillation border , where wave backs ( wb ) stop and wave fronts ( wf ) generate .based on an investigation of the dynamical nullcline and state variable , we concluded that a wave back will not pass a critical cell , , if the nullclines still intersect at the right branch when cell has dropped to its lower equilibrium ( fig .[ fig : nullclinemove](g , h ) ) .in contrast , if the intersections disappear , the state of will switch to a lower equilibrium .then cell is possible to oscillate , if it fires a wave front , in the case that the two nullclines do not cross at the left branch when the state of the inhibitor recovers to its lower limit ( fig .[ fig : nullclinemove](e , f ) ) .thus , we can roughly solve the condition by finding two possible tangency for the two nullclines eq .( [ eq : nullcline - v0 ] ) and eq .( [ eq : nullcline - u0 ] ) .this can be achieved using the following equations : equation ( [ eq : tangency ] ) is for two nullclines with the same slope . by substituting and into eq .( [ eq : tangency ] ) , we have from which we obtain two solutions   for cell to propagate a wave back , there should be only a lower equilibrium when close to the higher tangency point .the corresponding condition is .substitution leads to on the other hand , for cell to generate a wave front , there should be no lower equilibrium when is close to the lower tangency point .this simply means that , which can be rewritten as two critical conditions are shown in fig .[ fig:2con ] .    _ approximation _  1 . is the `` distal '' side of the critical cell .it remains in its higher equilibrium since the wave back can not pass it .thus , we can approximate it by finding the biggest intersection of the two nullclines . in the waveback case , since is close to the higher equilibrium , is nearly zero .thus , we determine to be 0.9 . in the wave front case , however , is small .if we consider the minus , we determine to be 0.8 . is the `` proximal '' side of the critical cell .it switches off before cell when a wave back comes and waits to be excited again by cell , so at the critical time , approaches 0 .figure [ fig : ut - vs - g ] shows according to eq .( [ eq : tangency - solutions ] ) .obviously , and change little , and can be regarded as the constants and , respectively .note that the above approximations are not valid when the coupling strength is too large , or excitability is too weak .with respect to . ]  ) representing the conditions of collective oscillation .a wave back passes the cell , and a wave front reflects , when the parameters are above the wb and wf line , respectively . ]according to the above approximations , by substituting into eq .( [ eq : cond - wb ] ) , and into eq .( [ eq : cond - wf ] ) , we obtain two rough conditions clearly , the critical coupling strength increases linearly with respect to the active factor .we draw two lines in fig . [fig : d - vs - g-2con ] , where the wb and wf lines are obtained from eq .( [ eq : simp - con - wb ] ) and eq .( [ eq : simp - con - wf ] ) , respectively .  in fig .[ fig : d - vs - g-2con ] , labels _c4 _ and _ c5 _ on the top horizontal axis indicate the value of defined by eq .( [ eq : gamma - profile ] ) for cells 3 , 4 and 5 , respectively . lines wf and wb cross each other between _ c4 _ and _ c5_. this kind of topology makes the 4 and 5 cells behave completely different .  for cell 5 , wf is above wb .if the coupling strength is between wf and wb , a wave back coming from the center can switch the 4 cell off , but the cell can not be switched on to fire a wave front .this is exactly the situation shown in fig .[ fig : beforeosc ] , in which no oscillation takes place . on the other hand , for cell 4 ,wf is below wb .clearly , if the coupling strength allows the wave back to suppress the 4 cell , the cell will be excited again and lead to a wave front .the conditions for these two situations depend on many other factors , such as the initial conditions , propagation velocity , cell excitability , and so on .the situation is much more complicated than the approximated case we have discussed here .qualitatively , we can conclude that the intersection of these two condition lines is the origin of self - sustained collective oscillation .in this paper , we have proposed a scenario for self - organized and self - sustained oscillation in multi - cellular biological tissue .in contrast to the usual framework based on an oscillatory genetic network , the present system does not include any self - oscillating cells .however , by mutual coupling , we can observe collective oscillation inside a group of cells , i.e. , tissue .moreover , oscillation can manifests itself in several ways , corresponding to different coupling strengths .anti - phase and in - phase oscillations at the two boundaries lead to changes in the position of oscillation and the oscillating cell population , respectively .the birth and death of oscillation resulting from variation in the coupling strength were also discussed .we also provide a general idea of how the size of the cell and population affects the oscillatory behavior . finally , a detailed investigation of the dynamical movement of the nullcline provided insight into the mechanism of complicated oscillatory phenomena .although there have been several studies on self - oscillatory phenomena in spatially discrete systems in the context of mathematics and physics , this paper extends these basic ideas to spatio - temporal self - organization in a biological system .it is of interest to extent our new hypothesis to spatial three dimensional systems , i.e. , a more realistic model of living organism .our observations were based on a numerical simulation .future analytical studies inspired by these interesting phenomenon are needed . atlast , but not less important , we are going to cooperate with biologists , in order to design corresponding biological experiments and to explore more proofs supporting our hypothesis .

Abstract: oscillations are ubiquitous phenomena in biological systems . conventional models of biological periodic oscillations usually invoke interconnecting transcriptional feedback loops . some specific proteins function as transcription factors , which in turn negatively regulate the expression of the genes that encode these `` clock proteins '' . these loops may lead to rhythmic changes in gene expression in a cell . in the case of multi - cellular tissue , collective oscillation is often due to synchronization of these cells , which manifest themselves as autonomous oscillators . in contrast , we propose here a different scenario for the occurrence of collective oscillation in a group of _ non - oscillatory _ cells . neither periodic external stimulation nor pacemaker cells with intrinsically oscillator are included in present system . by adopting a spatially inhomogeneous active factor , we observe and analyze a coupling - induced oscillation , inherent to the phenomenon of wave propagation due to intracellular communication .


secure communication over wireless fading channels becomes a critical issue due to the broadcast nature of wireless networks .traditionally , key - based cryptographic technologies  are used to secure the data transmission . however , the secrecy provided by cryptographic technologies is conditioned on the premise that the eavesdroppers have limited computational capability to decipher the message without the knowledge of secret keys .this premise becomes controversial with the rapid developments of computing devices . on the other hand ,physical layer security is an emerging research area that explores the possibility of achieving perfect - secrecy data transmission among legitimate network nodes , while possible malicious nodes eavesdropping the communication obtain zero information  . basically , the objective of physical layer security is to minimize the amount of confidential information that can be obtained by the illegitimate users according to their received signals . to achieve secure communications over wireless channels ,physical layer security explores time - varying properties of the fading channel , smartly designs the channel code , and processes the transmitted signals , instead of relying on encryption .the information - theoretic foundation of secret communication was laid by shannon in  .wyner s pioneering work introduced the wiretap channel model as a basic framework for physical layer security  , which was extended to broadcast channels with confidential messages described by csiszar and korner in  .these early works have led to a significant amount of recent research activities taking the fading characteristics of wireless channels into account .the basic system model of physical layer security over wireless channels is shown in fig . [fig : fig1 ] .most works in this area rely on the perfect knowledge of both the legitimate receiver s and eavesdropper s channels at the transmitter to enable secure encoding and advanced signaling. however , the assumption of perfect knowing channel state information ( csi ) is not realistic . in practical scenarios ,many reasons can cause imperfections in the csi at the communication nodes , such as the following ones :  * no feedback from the eavesdroppers . when the eavesdropper is a passive entity ,its csi or even location is very hard to be obtained at the legitimate transmitter .also , when the eavesdroppers are malicious users , they are not willing to provide their channel information to the legitimate party . * partial csi feedback from the receivers .the receivers sometimes can only provide partial csi feedback to the transmitter , e.g. , limited - rate feedback , channel direction feedback , and signal - to - noise ratio ( snr ) feedback . * imperfect feedback links between the transmitter and receivers .when the feedback links are not error - free or delayless , noise component is added into the feedback information , or the csi obtained at the transmitter is an outdated version of the actual csi . * channel estimation errors at the receivers .since the estimation of fading channels generally is not error - free , the csi obtained at the receivers is not perfect .  over the past few years , increasing attention has been paid to the impact of the uncertainty in the csi on both legitimate receiver s and eavesdropper s channels .the remainder of this paper is devoted to surveying and reviewing the literature of physical layer security with imperfect csi in wireless communications .we aim to provide a high - level overview of the current research and development of the field .in addition , this survey focus on the physical layer security researches without using the secret key , despite that some physical layer security work , e.g. ,  , investigated the secure transmission with the key observable by the eavesdropper over wireless channels .the remainder of this paper is organized as follows . for research from the information theoretic perspective , section [ sec : it ] presents the main performance metrics of secure transmissions with imperfect csi .signal processing secrecy enhancements developed in the secure transmission design considering imperfect csi are reviewed in section [ sec : sp ] .section [ sec : largesc ] turns to the secrecy with channel uncertainty in large - scale decentralized wireless .the open problem and possible future research directions are described in section [ sec : op ] .finally , section [ sec : conc ] concludes the paper .the performance limits of secure transmission systems with full csi are often characterized by the secrecy capacity .the secrecy capacity , , for degraded wiretap channel with additive gaussian noise was given by  , where and denote the shannon capacities of main ( legitimate receiver s ) and eavesdropper s channels , respectively .a positive secrecy capacity can be obtained only when the legitimate receiver s channel is better than the eavesdropper s channel . when considering fading channels , main and eavesdropper s channels for a specific fading realization can be regarded as complex additive white gaussian noise ( awgn ) channels .the shannon capacities of one realization of the quasi - static fading channels are given by  where and are the instantaneous snrs at the legitimate receiver and the eavesdropper , respectively .the instantaneous snr at the legitimate receiver is equal to , where denotes the transmit power , denotes the instantaneous channel gain at the legitimate receiver , and denotes the receiver noise variance at the legitimate receiver . also , the instantaneous snr at the eavesdropper is equal to , where denotes the instantaneous channel gain at the eavesdropper and denotes the receiver noise variance at the eavesdropper .thus , the secrecy capacity for one realization of the quasi - static fading channels can be written as  [ eq : cs_one ] c_s=  _ 2(1+_m)-_2(1+_e ) , _m>_e , + 0 ,   note that in order to achieve the secrecy capacity in eq .( [ eq : cs_one ] ) , the transmitter needs perfect knowledge of both and .  to measure the performance of secure transmissions over fading channels with imperfect csi , ergodic secrecy capacity and outage - based characterizationsare often adopted . inwhat follows , focusing on these two kinds of characterizations , we provide a review on the information theoretic aspect of the research in the field of physical layer security with imperfect csi . in addition , a brief description of the secrecy degrees of freedom , which applies for systems with pessimistic and strong csi assumptions , is presented as well .ergodic secrecy capacity applies for delay tolerant systems where the encoded messages are assumed to span sufficient channel realizations to capture the ergodic features of the channel .it captures the capacity limit under the constraint of perfect secrecy .typical examples of delay tolerant applications are document transmission and e - mail , which belong to the non - real time data traffic .gopala et al . presented ergodic secrecy capacity for both full csi case and the case of only main channel csi available at the transmitter .the secrecy capacity for one realization of the quasi - static fading channels is given in eq .( [ eq : cs_one ] ) .averaging over all fading realizations , the ergodic secrecy capacity of fading channels with full csi is given by where and are the distribution functions of and , respectively .since the transmitter has full csi on both channels , the transmitter can make sure that the transmission occurs only when .when only the channel gain of the legitimate receiver is known at the transmitter , the ergodic secrecy capacity is given by ^+\!f(\!\gamma_m\!)f(\!\gamma_e\!)\mathrm{d}\!\gamma_{\!m}\mathrm{d}\!\gamma_{\!e}\ ] ] in which ^+=\max\{x , 0\}$ ] .they outlined a variable - rate transmission scheme to show the achievability of ergodic secrecy capacity with only main channel information . during a coherence interval with the received snr at the legitimate receiver , the transmitter transmits codewords at rate .this variable - rate scheme relies on the assumption of large coherence intervals and ensures that when , the mutual information between the source and the eavesdropper is upper - bounded by . when , this mutual information is equal to .averaging over all the fading states , the achievable perfect secrecy rate is given by eq .( [ eq : erg_m ] ) .the secure message is hidden across different fading states .  .]  fig .[ fig : fig2 ] compares the ergodic secrecy capacity of the network with full csi at the transmitter and the ergodic secrecy capacity of the network with only main channel csi at the transmitter .the average channel qualities are , where is the expectation operation .the average power constraint is denoted by . according to eq .( [ eq : erg_f ] ) , the transmission occurs only when .thus , the constant power level used for transmission with full csi at the transmitter is , where denotes the probability measure .note that for , and hence . on the other hand ,the constant power level used for transmission with only main channel csi at the transmitter is .in addition , khisti and wornell studied the ergodic secrecy capacity for multiple - input , single - output , multiple - eavesdropper ( misome ) systems in  .they developed upper and lower bounds on the ergodic secrecy capacity with perfect csi on legitimate receiver s channel and imperfect csi on eavesdropper s channel .they also investigated the ergodic secrecy capacity of fasting fading channel for both high snr and finitely many antennas , i.e , the number of transmitted antenna is very large .rezki et al. studied the ergodic secrecy capacity for systems with imperfect csi on both legitimate receiver s and eavesdropper s channels at the transmitter . in , they presented a framework that characterizes the ergodic secrecy capacity of fast fading channels under imperfect legitimate receiver s channel estimation at the transmitter . in , they established upper and lower bounds on the ergodic secrecy capacity for single - input , single - output , single - eavesdropper ( sisose ) system with limited - rate feedback of the legitimate receiver s channel information .   as mentioned before , the ergodic secrecy capacity applies for delay tolerant systems which allow for the adoption of an ergodic version of fading channels .however , perfect secrecy can not always be achieved for systems with stringent delay constraints , and ergodic secrecy capacity is inappropriate to characterize the performance limits for such systems . on the other hand , outage - based characterizations , which measure systems with probabilistic formulations , become more appropriate . in  , assuming the fading is quasi - static , parada and blahut analyzed the scenario where the csi of both legitimate receiver s and eavesdropper s channels is not available at the transmitter .they provided an alternative definition of outage probability , wherein secure communications can be guaranteed for the fraction of time when legitimate receiver s channel is stronger than eavesdropper s channel .barros and rodrigues  firstly provided a detailed characterization of the outage secrecy capacity where the outage probability , , is characterized by the probability that a given target rate , , is greater than the difference between instantaneous main channel capacity , , and instantaneous eavesdropper s channel capacity , .the expression of is given by they also showed that fading alone guarantees that physical layer security is achievable , even when the eavesdropper has a better average snr than the legitimate receiver .in addition , bloch et al . characterized the relationship between the upper bound of outage probability and the variance of the channel estimation error on eavesdropper s channel in  .the secrecy outage behavior of a multiple - input , single - output , single - eavesdropper ( misose ) fading system is studied in  , where the authors proposed a relation between the degree of channel knowledge and the tolerable secrecy outage probability . in  , the outage - based formulations capture the probability of having a reliable and secure transmission .reliability and security are not distinguished , since an outage occurs whenever the transmission is either unreliable or not perfectly secure . in  ,zhou et al . presented an alternative secrecy outage formulation which directly measures the probability that a transmitted message fails to achieve perfect secrecy .the alternative secrecy outage is formulated by where and are the rate of transmitted codeword and the rate of the confidential information in the wire - tap code , respectively , and the outage probability is conditioned on a message actually being transmitted . from eq .( [ eq : pso ] ) , we see that the new formulation takes into account the system design parameters , such as the rate of transmitted codewords and the condition under which message transmissions take place .therefore , the alternative secrecy outage formulation is useful for the system designer to design transmission schemes that meet target security requirements .  .]  fig .[ fig : fig3 ] compares the secrecy outage probability of not having a reliable and secure transmission , in eq .( [ eq : pout ] ) , and the secrecy outage probability of not having a secure transmission , in eq .( [ eq : pso ] ) .the average received snrs are .the detailed discussion on the comparison of these secrecy outage probabilities can be found in  .apart from ergodic secrecy capacity and outage - based characterizations , another line of research , e.g. ,  , studied the performance limits of systems with imperfect csi under a pessimistic but strong assumption that allows the eavesdropper s channel to be arbitrarily varying .these work analyzed the so - called secrecy degrees of freedom ( s.d.o.f ) , which is the pre - log of the secrecy capacity at high snr and captures the asymptotic behavior of the achievable secrecy rate in high snr regime .the s.d.o.f . is formulated as where denotes the average power constraint on transmitted signals .the s.d.o.f region for single - user gaussian multi - input , multi - output ( mimo ) wiretap channel was investigated in  . the s.d.o.f .region of two user gaussian mimo broadcast channel with an arbitrarily varying eavesdropper channel was found in  .the s.d.o.f .region for two user gaussian mimo multiple access channel and gaussian two - way channel with the eavesdropper channel being arbitrarily varying were given in  and  , respectively .the work of  studied s.d.o.f .region for two - user mimo interference channel with an external eavesdropper .in addition , there is a main limitation of this type of works .it always requires some advantage in the antenna numbers on the legitimate receiver over the eavesdropper to get positive s.d.o.f ..in this section , we present various signal processing techniques for enhancing secrecy of wireless communications . specifically , secure on - offtransmissions for signal - antenna channels , beamforming with artificial noise for multi - antenna channels , and secure design techniques for relay channel and cognitive radio systems are described in the following three subsections .secure on - off transmission policy in wireless network designs generally works in the following way .the transmitter decides whether or not to transmit according to the knowledge of csi on the legitimate receiver s channel or eavesdropper s channel or both channels ( if applicable ) .transmissions take place whenever the estimated instantaneous csi fulfills the requirements related to some given thresholds , e.g. , snr thresholds .otherwise , transmissions are suspended .gopala et al . proposed a low - complexity on - off power allocation strategy according to the instantaneous csi on the legitimate receiver s channel , which approaches the optimal performance for asymptotically large average snr .zhou et al . designed two on - off transmission schemes , each of which guarantees a certain level of security whilst maximizing the throughput . with the statistics of eavesdropper s channel information , the first scheme requires csi feedback from the legitimate receiver to the transmitter , and the second scheme requires only 1-bit feedback . in  ,rezki et al . studied the system with imperfect legitimate receiver s csi and statistics of eavesdropper s channel at the transmitter .they derived the achievable rate of fast fading channel with a simple on - off scheme using a gaussian input .in addition , under various assumptions on the csi , he and zhou  proposed several secure on - off transmission schemes , which maximize the throughput subject to a constraint on secrecy outage probability .both fixed - rate and variable - rate transmissions were proposed . also , they not only considered the imperfect csi at the transmitter , but also studied the impact of imperfect csi at the receiver side .the work by hero  is arguably the first to consider secret communication in a multi - antenna transmission system , and sparked significant efforts to this problem  . for multi - antenna channels with imperfect csi ,beamforming with artificial noise is the one of the most widely - used techniques to secure the data transmission .negi and goel  first proposed an artificial noise injection strategy .in addition to transmit information signals , part of the transmission power is allocated to generate artificial noise in order to confuse the eavesdropper .specifically , the produced artificial noise lies in the null space of the legitimate receiver s channel , while the information signal is transmitted in the range space of the legitimate receiver s channel .this technique relies on the instantaneous csi on legitimate receiver s channel , but does not require the instantaneous csi on eavesdropper s channel .the legitimate receiver s channel nulls out the artificial noise .thus the legitimate receiver is not affected by the noise .the basic idea of beamforming with artificial noise is presented in fig .[ fig : fig4 ] .    in the following ,we illustrate the work after negi and goel s research , which studied the beamforming with artificial noise in various multi - antenna channel scenarios with different assumptions on the availability of csi .we first present the literature considering imperfect csi on the eavesdropper s channel , and then discuss the work which considers the imperfect csi on both eavesdropper s and legitimate receiver s channels .   the work in  studied the optimal power allocation between the information signal and the artificial noise for systems with both non - colluding and colluding eavesdroppers .the authors found that the equal power allocation is the strategy that achieves nearly the same secrecy rate as the optimal power allocation for the non - colluding eavesdropper case and more power should be used to transmit the artificial noise as the number of eavesdropper increases for the colluding eavesdropper case . in  , zhang et al . investigated the design of artificial - noise - aided secure multi - antenna transmission in slow fading channels .they provided throughput - maximizing design solutions with either fixed - rate or adaptive - rate encoder , both including the optimal rate parameters of the wiretap code as well as the wise transmit power allocation between the information signal and the artificial noise .huang and swindlehurst  obtained the robust transmit covariance matrices on worst - case secrecy rate maximization under both individual and global power constraints .they investigated both cases of direct transmission and cooperative jamming with a helper .in addition , gerbracht et al . characterized the optimal single - stream beamforming with the use of artificial noise to minimize the outage probability .they pointed out that the solution converges to maximum ratio transmission ( mrt ) for the case of no csi to the eavesdropper , and the optimal beamforming vector converges to the generalized eigenvector solution with the growing level of csi .lin et al . showed that the artificial noise selected in  is suboptimal . according to their study , the eigenvectors of the optimal covariance matrices of both information signals and generalized artificial noise are equal to the right singular vectors of the legitimate receiver s channel , and the power of artificial noiseshould be allocated uniformly over the eigenvectors .the rigorous proofs for these facts were also provided .  in the work of  , although the instantaneous csi on the eavesdropper s channel is not required , the transmitter still needs the statistics of eavesdropper s channel . for the case where no csi on the eavesdropper s channel ( including the statistics ) is known at the transmitter , swindlehurst andmukherjee  proposed a modified water - filling algorithm which balances the required transmit power with the number of spatial dimensions available for jamming the eavesdropper .as described in the modified water - filling algorithm , the transmitter first allocates enough power to meet a target performance criterion , e.g. , snr or rate , at the receiver , and then use the remaining power for broadcasting artificial noise .they also applied the similar algorithm to investigate the multiuser downlink channels in  .the imperfect csi on the legitimate receiver s channel at the transmitter mainly incurs two problems .first , without knowing the actual instantaneous csi on the legitimate receiver s channel , the transmitter can not make sure that the data transmission rate is not larger than the legitimate receiver s channel capacity .then , a transmitted packet is unable to be decoded by the receiver , i.e. , the packet is corrupted , whenever the data transmission rate exceeds the legitimate receiver s channel capacity .second , with imperfect instantaneous csi on the legitimate receiver s channel , the artificial noise leaks into the legitimate receiver s channel , since the beamforming with artificial noise is designed according to the estimated instantaneous csi rather than the actual instantaneous csi .therefore , the artificial noise interferes the desired user , despite that it is intended to only confuse the eavesdropper .naturally , the techniques applying for systems with perfect csi on the legitimate receiver s channel become not optimal .taylor et al . presented the impact of the legitimate receiver s channel estimation error on the performance of an eigenvector - based jamming technique in .their research showed that the ergodic secrecy rate provided by the jamming technique decreases rapidly as the channel estimation error increases .mukherjee and swindlehurst  also pointed out that the security provided by beamforming approaches is quite sensitive to the imprecise channel estimates .they proposed a robust beamforming scheme for mimo secure transmission systems with imperfect csi of the legitimate receiver .pei et al . addressed a stochastic time - varying csi uncertainty model with uplink - downlink reciprocity . using this model, they proposed a new iterative algorithm to secure the transmission , which is robust to csi errors .the authors of  investigated the effects of imperfect csi on the optimal power allocation and the critical snr for secure communications .they found that allocating power on the artificial noise to confuse the eavesdropper is better than increasing the signal strength for the legitimate receiver as the channel estimation error increases . adapting the secrecy beamforming scheme , liu et al . investigated the joint design of training and data transmission signals for wiretap channels . the ergodic secrecy rate for systems with imperfect channel estimations at both the legitimate receiver and the eavesdropper was derived .the secrecy rate is hard to be calculated , since the channel estimation errors cause non - gaussianity of equivalent noise .they solved this problem by the large number of transmit - antenna analysis .based on the achievable ergodic secrecy rate , they found the optimal tradeoff between the power used for training and data signals .furthermore , advocating the joint optimization of the transmit weights and artificial noise spatial distribution with the use of a quality - of - service ( qos)-based perspective , liao et al . proposed a secret - transmit beamforming approach in accordance with the imperfect csi on both the legitimate receiver s and eavesdropper s channels .in  , ng et al . addressed a resource - allocation and scheduling optimization problem for orthogonal frequency division multiple access ( ofdma ) networks .the optimization problem takes into account artificial noise generation and the effects of imperfect csi in slow fading .they proposed a resource allocation algorithm considering secrecy outage , channel outage , and the potentially detrimental effect of artificial noise generation . considering the systems with partial csi feedback , lin et al .investigated the scenario where only quantized channel direction information ( cdi ) of the legitimate receiver s channel is available at the transmitter in  . for a given transmission power and a fixed number of feedback bits , they derived the optimal power allocation among the information signal and the artificial noise to maximize the secrecy rate under artificial noise leakage .in addition , it is necessary to mention that there also exist some studies not considering the artificial noise to enhance the security of multi - antenna systems with imperfect csi .some examples , i.e. ,  , of these studies are illustrated as follows . in  , li and petropulusolved optimal input covariance that maximizes the ergodic secrecy rate subject to a power constraint for misose systems with imperfect csi on eavesdropper s channel .li and ma  formulated a transmit - covariance optimization problem for secrecy - rate maximization ( srm ) of misome systems with imperfect csi on both main and eavesdropper s channels .the authors of  analyzed the systems with only statistics of main and eavesdropper s channels at the transmitter .they showed that the secure beamforming is still secrecy capacity achieving in such a scenario , and proposed the optimal channel input covariance matrix , which fully characterizes the secrecy capacity .they also pointed out that the artificial noise is not necessary in this case .geraci et al .studied secrecy sum - rates achievable by regularized channel inversion ( rci ) precoding in miso systems under imperfect csi in  .secure communication assisted by relay nodes is often regarded as a natural extension to the secure transmission in multi - antenna networks .the physical layer security can be provided by careful signaling at different relays in the system .a virtual beam towards the legitimate receiver can be built by the collaboratively work among relay nodes , which is similar to the secure transmission in multi - antenna systems . however , unlike the multiple - antenna transmission , the transmitter can not directly control the relays . for the network of single - antenna wiretap channel with serval relays , goel and negi described a 2-phase protocol to obtain coordination in transmitting artificial noise among the relays .in the first phase , the transmitter and the legitimate receiver both transmit independent artificial noise signals to the relays .different linear combinations of these two signals are received by the relays and the eavesdropper . in the second phase, the relays replay a weighted version of the received signal , using a publicly available sequence of weights .meanwhile , the transmitter transmits the confidential information , along with a weighted version of the artificial noise transmitted in the first stage . with the knowledge of the artificial noise component due to the legitimate receiver , the legitimate receiver is able to cancel off the artificial noise and get the confidential information . assuming global full csi at every node , the work in  provided a detailed analysis on secure communications of one source - destination pair with the help of multiple cooperating relays in the presence of one or more eavesdroppers , which includesdecode - and - forward ( df ) , amplify - and - forward ( af ) , and cooperative jamming ( cj ) three different cooperate schemes .  to explore the effects of imperfect csi in relay systems , researchers often consider the uncertainty of csi on three kinds of links , which are relay - destination links , relay - eavesdropper links , and source - relay links .the authors of  investigated the effect of imperfect csi on the relay - eavesdropper channels .they proposed a df relaying protocols for secure communication , which maximizes the lower bound on the ergodic secrecy capacity under a total relay transmission power constraint . considering the imperfect csi on the channels from relay to destination and relay to eavesdropper , zhang and gursoy  provided optimization frameworks for the robust df - based relay beamforming design .furthermore , vishwakarma and chockalingam  computed the worst case secrecy rate when there are imperfections in the csi on all the links , i.e. , relay - destination links , relay - eavesdropper links , and source - relay links .cognitive radio ( cr ) has been widely recognized as an effective technology to improve the utilization of wireless spectrum by allowing secondary users to coexist with primary users and access the spectrum of the prime system . in cr systems with secrecy message broadcasted in the primary links , the signals of secondary links for their users can also serve as the artificial noise for the secure primary link . in order to confuse the eavesdropper overhearing the primary links , the secondary system operates similar to that of the helper nodes , but simultaneously serving their own receivers .the papers that study the imperfect csi in such cognitive radio systems can be found in  and references within .the authors of explored miso cr systems where the secondary system secures the primary communication in return for permission to use the spectrum . on the other hand , when the secondary user transmitter sends confidential information to a secondary user receiver on the same frequency band with a primary user , the requirement of not interfering the primary users is often treated as a power constraint on the transmitted signals in the secondary system . assuming all csi is imperfect known , pei et .al  explored the optimal secondary user transmitter design , which maximizes the secure transmission rate of the secondary link while avoiding harmful interference to the primary users .they proposed two approaches to solve this challenging optimization problem , which is non - convex and semi - infinite .in the last section , we summarized the research results on physical layer security enhancements for systems consisting of a small number of nodes .we now turn our attention to another very important class of wireless networks : large - scale decentralized wireless networks . in such networks ,the csi of eavesdroppers is rarely available at legitimate users .even the locations of eavesdroppers may not be known .the lack of eavesdropper s information makes communication security a challenging problem . apart from that , the decentralized nature of the network rules out any global optimization approach for secrecy enhancements .pioneering works on physical layer security in large - scale decentralized wireless networks focused on the connectivity analysis .specifically , the notion of secrecy graph was introduced in  and further developed in  to include fading channels .various secure connectivity improvements were discussed in  , including multi - antenna sectoring and beamforming .the connectivity in the presence of the location uncertainty of eavesdroppers was studied in  .building up from the connectivity analysis of the secrecy graph , the secrecy capacity scaling was analyzed in  .specifically , the studies in  showed that the secrecy requirement does not reduce the capacity scaling of the network , i.e. , the capacity scaling law is the same for both insecure message transmission and secure message transmission .of course , achieving such an optimal scaling law under the secrecy constraint requires very different transmission and access protocols .for example , when eavesdroppers locations are unknown , various secrecy enhancements such as cooperative jamming and multi - path transmission in conjunction with network coding may be required  .although the scaling law results may provide insights into the asymptotic secrecy throughput performance of large - scale networks , a finer view of throughput is necessary to better understand the impact of key system parameters and transmission protocols , since most of design choices affect the actual ( non - asymptotic ) throughput but not the scaling behaviors . to this end, a new performance metric named secrecy transmission capacity was developed in  to capture the area spectral efficiency of secure transmission .the formulation of such a metric was based on the outage approach in  which accommodates the practical scenario where the csi of the eavesdroppers is unknown to the legitimate nodes .despite the increasing attention paid to the effect of imperfect csi on the physical layer security , research on this area is still at an early stage . in this section ,we discuss some open problems in the research area of the physical layer security with imperfect csi .   among the existing work on physical layer security considering imperfect csi ,most investigates the impact of imperfect csi at the transmitter while assuming perfect channel estimation at the receivers .however , only a limited amount of work , e.g. , , paid attention to the imperfect channel estimation at the receivers .clearly , the assumption of perfect channel estimation at the receiver is not very practical , since the estimation of fading channels generally is not error - free . in principle, the channel estimation error exists at both the legitimate receiver and the eavesdropper .assuming perfect estimation at the eavesdropper is more reasonable from the secure transmission design point of view , since it is often difficult or impossible for the transmitter to know the accuracy of the eavesdropper s channel estimate .nevertheless , in scenarios where the eavesdropper is just an ordinary user of the network whose performance and other information can be tracked by the transmitter , e.g. , , the consideration of imperfect channel estimation at the eavesdropper becomes relevant .   with few exceptions , almost all the existing research studies on physical layer security assumed that the eavesdropper s location is perfectly known .the validity of such an assumption strongly depends on the application under investigation .for example , when the eavesdropper is a passive entity without transmission , its location is very hard to be obtained . also , in large - scale complex networks , it is very difficult to obtain eavesdroppers locations due to the random deployments or mobility of nodes. therefore , it is an interesting research direction to consider the secrecy in networks with imperfect or no knowledge of the eavesdropper s location at the transmitter .in this paper , we reviewed the research in physical layer security with practical assumptions on fading channel information .for the characterizations of performance limits , we described ergodic secrecy capacity suitable for delay tolerant systems , outage - based characterizations for systems with stringent delay constraints , and secrecy degrees of freedom . also , we surveyed the signal processing secrecy enhancements proposed for different transmission scenarios , i.e. , secure on - off transmission for signal - antenna channels , beamforming with artificial noise for multi - antenna channels , and other signaling designs for relay channels or cognitive radio systems .in addition , the recent results on secrecy in large - scale decentralized wireless networks were reviewed .future research directions on physical layer security with imperfect csi include the imperfect channel estimation at receiver sides and the imperfect knowledge of the eavesdropper s location .z. rezki , a. khisti , and m .- s .alouini , `` on the ergodic secrecy capacity of the wiretap channel under imperfect main channel estimation , '' in _ proc .asilomar conf .signals syst ._ , pacific grove , ca , nov .2011 , pp . 952957 .x. zhou , m. r. mckay , b. maham , and a. hjrungnes , `` rethinking the secrecy outage formulation : a secure transmission design perspective , '' _ ieee commun ._ , vol . 15 , no . 3 , pp . 302304 , mar. 2011 .x. he and a. yener , `` mimo wiretap channels with arbitrarily varying eavesdropper channel states , '' submitted to_ ieee trans .inf . theory _, july 2010 .[ online ] .available : http://arxiv.org/abs/1007.4801 .b. he and x. zhou , `` secure on - off transmission design with channel estimation errors , '' submitted to _ ieee trans .forensics security _ , apr[ online ] .available : http://arxiv.org/abs/1304.6485   a. mukherjee , s. a. a. fakoorian , j. huang , and a. l. swindlehurst , `` principle of physical layer security in multiuser wireless networks : a survey , '' nov . 2010 .[ online ] .available : http://arxiv.org/abs/1011.3754    x. zhou and m. r. mckay , `` secure transmission with artificial noise over fading channels : achievable rate and optimal power allocation , '' _ ieee trans ._ , vol .59 , no . 8 , pp . 38313842 , oct .x. zhang , x. zhou , and m. r. mckay , `` on the design of artificial - noise - aided secure multi - antenna transmission in slow fading channels , '' to appear in _ ieee transtechnol . _ ,[ online ] .available : http : http://arxiv.org / abs/1212.3640   p .- h .lin , s .- c .lin , s .- h .lai , and h .- j .su , `` on secure beamforming for wiretap channels with partial channel state information at the transmitter , '' in _ proc .apsipa asc _ ,hollywood , ca , dec 2012 , pp .15 .j. m. taylor , m. hempel , h. sharif , s. ma , and y. yang , `` impact of channel estimation errors on effectiveness of eigenvector - based jamming for physical layer security in wireless networks , '' in _ proc .ieee camad workshop _ , kyoto , japan , june 2011 , pp. 122126 .liao , t .- h .chang , w .- k .ma , and c .- y .chi , `` qos - based transmit beamforming in the presence of eavesdroppers : an optimized artificial - noise - aided approach , '' _ ieee trans . signal process ._ , vol .59 , no . 3 , pp . 12021216 , mar .2011 .lin , t .- h .chang , y .- l .liang , y .- w .p. hong , and c .- y .chi , `` on the impact of quantized channel feedback in guaranteeing secrecy with artificial noise : the noise leakage problem , '' _ ieee trans .wireless commun ._ , vol . 10 , no . 3 , pp .901915 , mar .2011 .g. geraci , r. couillet , j. yuan , m. debbah , and i. b. collings , `` secrecy sum - rates with regularized channel inversion precoding under imperfect csi at the transmitter , '' in _ proc .ieee icassp _ , may 2013 .y. pei , y .- c .liang , k. c. teh , and k. h. li , `` secure communication in multiantenna cognitive radio networks with imperfect channel state information , '' _ ieee trans .signal process ._ , vol .59 , no . 4 , pp .16831693 , apr .2011 .p. c. pinto , j. barros , and m. z. win , `` secure communication in stochastic wireless networks part ii : maximum rate and collusion , '' _ ieee trans .forensics security _ , vol . 7 , no . 1 ,139147 , feb .2012 .      c. capar , d. goeckel , b. liu , and d. towsley , `` secret communication in large wireless networks without eavesdropper location information , '' in _ proc .ieee infocom _, orlando , fl , mar .2012 , pp . 11521160 .x. zhou , r. k. ganti , j. g. andrews , and a. hjrungnes , `` on the throughput cost of physical layer security in decentralized wireless networks , '' _ ieee trans .wireless commun ._ , vol . 10 , no . 8 , pp .27642775 , aug .2011 .    c. y. leow , .apar , d. goeckel , and k. k. leung , `` two - 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Abstract: physical layer security is an emerging technique to improve the wireless communication security , which is widely regarded as a complement to cryptographic technologies . to design physical layer security techniques under practical scenarios , the uncertainty and imperfections in the channel knowledge need to be taken into consideration . this paper provides a survey of recent research and development in physical layer security considering the imperfect channel state information ( csi ) at communication nodes . we first present an overview of the main information - theoretic measures of the secrecy performance with imperfect csi . then , we describe several signal processing enhancements in secure transmission designs , such as secure on - off transmission , beamforming with artificial noise , and secure communication assisted by relay nodes or in cognitive radio systems . the recent studies of physical layer security in large - scale decentralized wireless networks are also summarized . finally , the open problems for the on - going and future research are discussed .  physical layer security , fading channels , channel uncertainty , imperfect channel state information .